MASARYK UNIVERSITY Faculty of Science Department of Mathematics and Statistics Riccati methods for half-linear differential equations Habilitation Thesis Zuzana Pätikovä Brno 2021 Abstract The form of the thesis is a commentary on a collection of eight scholarly papers devoted to studies of the second-order half-linear differential equations. We are interested mainly in the qualitative theory and asymptotic properties of studied equations, focusing on the applicability of the so-called modified Riccati technique. With its use, it is possible to investigate especially conditionally oscillatory equations, which occur at the threshold between oscillation and nonoscillation, and their perturbations. After the introductory part presenting the historical background and the methodology, the results providing concrete asymptotic formulas are commented. The next sections sum up our results in the form of various oscillation and nonoscillation criteria for ordinary and neutral half-linear equations. In the end, the numerical approach to finding approximate solutions of half-linear Euler-type equations, which makes use of the differential transformation method, is introduced. The thesis is concluded with a list of possible future directions of research. Contents Preface .................................................................... 1 Chapter 1. Introduction ................................................... 2 1.1 Historical background..................................... 3 1.2 Methods in qualitative theory of half-linear equations............... 7 Chapter 2. Asymptotic formulas of nonoscillatory solutions .................. 14 2.1 Nonoscillatory solutions of conditionally oscillatory half-linear equations . 15 2.2 Subcritical half-linear Euler-type equation....................... 17 Chapter 3. Oscillation and nonoscillation criteria ........................... 18 3.1 Hille-Nehari type criteria .................................. 18 3.2 Hille-Wintner type integral comparison theorems.................. 23 Chapter 4. Neutral half-linear equations .................................... 26 4.1 Oscillation criteria....................................... 28 4.2 Application to neutral Euler-type equations...................... 29 Chapter 5. Numerical approach ............................................ 34 5.1 Differential transformation method............................ 34 5.2 Application of the differential transformation to half-linear Euler equation . 36 Conclusion ................................................................. 41 — Hi— Preface Since establishing the foundations of calculus, differential equations are one of the most widely used tools for describing continuous real processes. The object of our interest -half-linear second-order differential equations - can be seen either as a generalization of ordinary linear second-order differential equations or as a scalar case of partial differential equations with p-Laplacian. The qualitative theory of half-linear equations has been deeply developed during the last decades and its research still continues. The thesis is based on the papers [65, 66, 67, 109, 110, 111, 112, 114]. In the body of the thesis, their results are summarized and commented, and printouts of the full versions are enclosed at the end in the form of an attachment. The work is organized as follows. In the first chapter, we provide a brief introduction to the topic in the historical context and present the main proving techniques used in our research. The second chapter is devoted to studies of asymptotic formulas for nonoscillatory solutions of the so-called conditionally oscillatory half-linear differential equations ([109, 110]). The third chapter brings a variety of oscillation results in the form of integral oscillation and nonoscillation criteria ([65, 66, 111]). An extension of some oscillation criteria to neutral half-linear differential equations is presented in the fourth chapter ([67]). In the last chapter, we change our perspective and turn our attention to numerical methods for finding approximate solutions of half-linear second-order differential equations, in particular to application of the differential transformation method to Euler type half-linear equations without delay and with proportional delay ([112, 114]). I would like to express my thanks to my coauthors for their fruitful cooperation and for motivation. My greatest gratitude goes in memoriam to Professor Ondřej Došlý for his kindness, willingness, patience, guidance, and inspiration. -1- Chapter 1 Introduction The object of study of the thesis is mainly the half-linear second-order differential equation of the form (r(t) 1, and r, c are continuous functions, r(t) > 0. For p = 2 equation (1.1) reduces to the second-order linear Sturm-Liouville differential equation (also called second-order homogenous self-adjoint equation) (r(t)x;Y + c(t)x = 0. (1.2) From this point of view, the study of the properties of its generalization (1.1) is a natural direction of research. On the other hand, half-linear equation (1.1) can be achieved as a transformation of some partial differential equations with the so-called p-Laplacian Apu:= div(||Vw|p-2Vw), where for u(x) = u(x\,... ,xn), N e N, the symbol Vu stands for the Hamilton nabla operator and div denotes the usual divergence operator. If u is a radially symmetric function and t = \ \x\\, then the transformation u(x) = y(t) reduces the partial differential p-Laplacian operator to the ordinary differential operator Apu(x)=t'-N(t^(y'(t))y, ' = ± In this connection half-linear equation (1.1) with r(t) = 1 can be considered as a class of differential equations with the one-dimensional p-Laplacian. Origins of p-Laplacian are described, for example, in the paper [4]. Accordingly, the history of p-Laplacian is closely linked to applications in the filtration of fluids through porous media and nonlinear non-Newtonian fluid dynamics. Another application can be found, for example, in [2]. Here p-Laplacian is used to model a non-homogenous diffusion to determine the height of a growing pile of noncohesive sand, where an ordinary differential equation arises in the limit case of "infinitely fast/slow" diffusion (see also [107]). Furthermore, half-linear equation (1.1) appears as a special case of the quasilinear equation of the form (r(t)Pl(x'))' + c(t)P2(x)=0, ®Pl(x) = M^sgnx, 1 and the coefficient functions r, c satisfy the same assumptions as in equation (1.1). If p\ = P2, then equation (1.3) reduces to equation (1.1). In this context, half-linear equation (1.1) can be understood as the borderline between the so-called super-half-linear and sub-half-linear equations, when p\ > p2 and p\ < p2, respectively. In the qualitative theory of quasilinear equation (1.3), half-linear equation (1.1) can be used for comparison purposes. Finally, the quasilinear equation (1.3) can be generalized to the following second-order nonlinear equation of the form (r(t) 0 and r(ř),c(ř) are continuous, the existence and uniqueness theorem for an initial value problem can be proved. Hence solutions of (1.1) are extendable for arbitrarily large intervals. In the following years, it was found out that the oscillation theory of half-linear equation (1.1) evinces many very similar aspects as that of linear equation (1.2). For example, the half-linear version of Sturmian theory extends from the linear theory almost verbatim. However, the missing additivity of the solution space of half-linear equations also causes some differences, as, for example, the unavailable Wronskian identity and Fredholm alternative. These missing tools have to be overcome by new approaches, which, surprisingly, sometimes lead to ways to obtain results that are new even in the linear case. In the last decades of the 20th century and at the beginning of the new millennium, the qualitative theory of half-linear equations was thoroughly developed by many authors. A partial survey of the results was made in a chapter of the book [1] by Agarwal, Grace, and O'Regan in 2002. Three years later, in the year 2005, the monograph by Došlý and Řehák [39] was published, and since then, it has been widely used as a reference book by many authors. The book exhaustively sums up the basics of the half-linear theory, as Chapter 1. Introduction 4 well as many branches of relevant research up to the year of publication. After 2005, work on the research of half-linear equations has naturally continued. During this period, until today, hundreds of papers dealing with equation (1.1) and with equations or their systems generalizing it were published. Due to the content of this thesis, we consider mainly ordinary differential equations. Concerning partial differential equations related to half-linear equation (1.1), we refer, for example, to the group of authors around professor Drábek from Pilsen (see, for example [52, 53]). In the rest of the section, we try to indicate the main directions that the study of equation (1.1) since 2005 went on. This attempt is not meant as an exhaustive overview, which exceeds the possibilities of this work. The aim is to show the main branches of the current research of half-linear equations, also that the topic is still actual and that it is being addressed by scientists all around the world. The Japanese school of half-linear differential equations has played an important role in the research of half-linear equations throughout their whole history. The leading personality of the group working in the theory of half-linear equations in the observed period has been professor Sugie. Since 2005, he has collaborated on publications related to half-linear equations with many coauthors, for example, Yamaoka, Onitsuka, Yamaguchi, Kono, Mat-sumura, Hata, Wu, Ishibashi, and others. Sugie's interest has been not only in half-linear equations of the form (1.1), but also in other related fields such as nonlinear differential equations with p-Laplacian, delayed half-linear equations, nonautonomous half-linear differential systems and later also damped half-linear oscillators. From their papers, let us refer at least to some of those dealing with equation (1.1): [117, 118, 119, 122]. Very recently, the asymptotic properties of solutions of (1.1) were studied by Naito in [106]. Different groups of coauthors published their papers on half-linear differential systems and damped half-linear oscillators (Onitsuka, Soeda, Naito, Pasic, Tanaka, Enaka). The interest of Yoshida was in elliptic equations and inequalities with p-Laplacian. Yamaoka in [123] considered nonlinear perturbations of the half-linear Euler equation. Recently, a very new research direction has been introduced by Yamaoka and Fujimoto, who admit p to be a function p(t) ([68, 69]). Next, let us focus on the links between the two schools in Japan and in Brno. In addition to the collaboration between Yamaoka and Došlý, and a very recent common work of Fujimoto and Došlá, let us mention the position of professor Jaroš from Bratislava. In his publications with Brno coauthors ([34, 94]) he proved some oscillation and nonoscillation criteria for (1.1). In his own papers, he focused mainly on comparison and integral criteria for even-order half-linear equations ([85, 86, 87]), and he presented a reduction-of-order approach for higher-order equations ([88]). In cooperation with Kusano and Tanigawa, several papers were published not only for equation (1.1) ([91]) but also for half-linear differential systems ([89, 92]) and the fourth-order equation ([93]). Another frequent coauthor of Kusano and Tanigawa has been Manojlovič. Their cooperation resulted in papers on the existence of regularly varying solutions for retarded half-linear equations ([96,100]), on criteria for the fourth-order half-linear equations ([97]) and asymptotics of solutions of (1.1) ([95]). With respect to the author's scientific background and roots, we further focus on a more detailed description of the role of the Brno school. In the observed period, Brno has been one of the main research centers, where a group of authors around the leading personality of Chapter 1. Introduction 5 professor Došlý has continued in working on the theory of half-linear equations. Professor Došlý published many papers on equation (1.1) either on his own, with his Ph.D. students (e.g., Řezníčková, Fišnarová, Pátíková, Hasil, Veselý), or other coauthors (e.g., Lomtatidze, Ünal, Bognar, Özbekler, Cecchi, Marini, Yamaoka, Jaroš). The contribution of Došlý was in his innovative approach and many original ideas leading to new branches of the research field. To present some of the topics of his studies from this period in more detail, let us mention the following ones: • various (non)oscillation criteria ([25, 35, 38, 41, 42, 46, 50, 37]), • half-linear Euler type equation with its generalizations and perturbations ([19, 20, 30, 32, 40]), • general conditionally oscillatory equations ([34, 47]), • minimal solution of Riccati type equation ([8]), • asymptotics of solutions ([9, 43, 48, 21]), • equations with periodic coefficients and coefficients having mean values ([31, 33, 36, 49]), • involving perturbation in both the terms of equation (1.1) ([23, 24, 26]), • power comparison theorems ([28]), • distinguishing the principal solution and its characterization ([15, 17]), • higher-order half-linear differential equations ([44, 45]). The most frequent methods used by Došlý include the Riccati technique and its modification, which is referred to as the modified Riccati technique, further comparing half-linear equations with linear equations, variational technique, and the Prüfer angle technique. The untimely decease of Došlý was a great loss for the entire scientific community working in the theory of half-linear differential equations. The second author of the monograph [39] Řehák also continued in his work, especially in studies of asymptotics of solutions of half-linear equations in the frame of regularly varying functions. We refer to the survey paper [115], where a significant part of his previous work is summed up and where the classification of solutions in the sense of regular variation is finalized. Řehák extended oscillation and asymptotic theory also to half-linear difference and dynamic equations. To the mental members of the Brno school also belongs the pair of coauthors Mařík and Fišnarová. Apart from their joint cooperation with Došlý ([27, 29]), let us mention their work on extending the Picone's identity ([59]), presenting a new class of constants for which Hille-Nehari type criteria hold ([61]), working with the modified Riccati technique and discussing the integral characterization of principal solution ([60, 62]). In the last years, their joint research has shifted to half-linear equations with delay and to neutral half-linear equations (see, for example [63, 64]). Mařík published several papers in fields linking the half-linear ordinary differential equations with partial differential equations with p-Laplacian (e.g., [102, 103]). Fišnarová has cooperated on a few papers also with Pátíková, who is the author of this thesis. Pátíková has worked mainly with perturbations of conditionally oscillatory and Euler-type equations ([65, 66, 111]), proved some asymptotic formulas for nonoscillatory solutions ([108, 109, 110]), considered neutral half-linear Chapter 1. Introduction 6 equations ([67]) and worked within the numerical approach on the application of the differential transformation method to half-linear Euler-type equations ([112]). In the last years, a significant contribution to the topics studied in Brno has been made by the pair of coauthors Hasil and Veselý. They master and contribute mainly to techniques based on the generalized Priifed transformation, averaging technique, and adapted Riccati transformation. Their interest has been in half-linear equations with periodic and almost periodic coefficients ([75, 76, 77, 78, 80, 83]) and coefficients having mean values ([73, 82]), they studied conditional oscillation of half-linear equations and of Euler-type equations ([18, 71, 79, 81]). Hasil and Veselý have extended some of their results also to half-linear equations on time scales and half-linear difference equations. Recently, they collaborate also with Ph.D. students (Juránek, Sišoláková, [72, 74]). To complete the overview of the Brno school, let us also mention Sremr, who with his coauthors published results for systems of half-linear differential equations (see, for example [51]), Lomtatidze, whose interest was, for example, in boundary value problems, and the three collaborating authors Cecchi, Došlá and Marini. At the beginning of the regarded period, in the work of the three, they considered equation (1.1) and presented some results on the principal solution and asymptotic properties of solutions ([13, 14, 16]). Later, their attention has turned to more general nonlinear equations, for which half-linear equations are a special case. In the last years, delayed, advanced, and neutral half-linear equations have attracted considerable attention. Apart from the authors already mentioned, let us present at least the strongest groups of coauthors from all over the world: Bohner, Grace, Jadlovská, Džurina, Li, Chatzarakis, Stavroulakis, Baculíková, Graef, Li, Zhang, Santra, Bazighifan (see, for example, the latest results and references therein [10, 11, 54, 116]). Concerning other topics, which belong to the half-linear theory but which are less relevant to our work, let us continue with only a brief sketch of other branches. Half-linear equations also appear as a special class of some nonlinear/functional equations (see, for example, the works of Tiryaki). For the half-linear impulsive differential equations, see the papers of Zafer, Ózbekler, and their coauthors. Tiryaki, Zafer, and Ózbekler have also studied Lyapunov-type inequalities with the use in the theory of various types of half-linear equations (see also papers by Liu, Dhar, Kayar, Agarwal, Kong, Cakmak). Sahinez, Zafer, and Tiryaki also worked on elliptic equations with p-Laplacian. Half-linear dynamic equations on time scales have been studied by a large number of authors (for example, Saker, Agarwal, Peterson, Zafer, Zhang, Bohner, Hassan, Li), as well as half-linear difference equations. From the previous overview, it can be seen that the theory of half-linear equations is very rich, diverse, and deeply developed. The aim of this work is to present the contribution of the author to this theory. In particular, we have focused on the qualitative behaviour and asymptotical properties of solutions of (1.1) on a neighbourhood of infinity, i.e., for t e [T,°°). We use the modified Riccati technique to obtain various types of oscillation and nonoscillation criteria, which are used mainly for equations at the boundary between oscillation and nonoscillation. Furthermore, we introduce the obtained asymptotic formulas of nonoscillatory solutions of certain equations, present our oscillatory criteria for neutral equations, and introduce the use of the differential transformation in the search for a numerical solution of the half-linear Euler equation. Chapter 1. Introduction 7 1.2 Methods in qualitative theory of half-linear equations In this section, we recall relevant preliminaries, remind the concept of principal solution, summarize some knowledge about conditionally oscillatory equations and Euler-type equations, and state the basis of the proving technique. Most of the preparatory work leads to the introduction and implementation of the so called modified Riccati technique, which plays key role in our research. Recall that the core idea of the modified Riccati technique was introduced by Došlý at the end of the last century and although in his older papers the technique was not named explicitly, its variations are present in many of his publications. The classification of solutions of the object of our study - equation (1.1) - in terms of oscillation is used in the same manner as in the linear theory. If a solution is of one sign in some neighbourhood of infinity, it is called nonoscillatory. Otherwise, i.e., when a nontrivial solution has got infinitely many zeros tending to infinity, it is called oscillatory. Half-linear equation (1.1) can be classified as oscillatory if all its solutions are oscillatory and nonoscillatory if all of them are nonoscillatory. The fact that if equation (1.1) has one non(oscillatory) solution, then the equation preserves the same non(oscillation) property, is a direct consequence of the half-linear version of Sturm separation theorem (see, for example, [39, Theorem 1.2.3]). Hence, oscillatory and nonoscillatory solutions of (1.1) cannot coexist and oscillation or nonoscillation of equation (1.1) is implied by the form and properties of the coefficient functions r(t) and c(t). If oscillatory behaviour, roughly speaking, depends only on a constant, we talk about the so-called conditional oscillation. More precisely, assuming in addition that c(t) is positive and 7 is a positive real parameter, the equation (r(t) 70 and nonoscillatory for 7 < 7o- The number 70 is then called an oscillation constant of equation (1.4). The main tools of proving techniques used in the half-linear theory make use of the Prüfer transformation or are implied by the so-called Roundabout theorem (see, for example, [39, Section 1.1.3] and [39, Theorem 1.2.7], respectively). The Roundabout theorem presents relationship between solvability of Riccati equation, positivity of the p-degree functional F(y,a,b)= [b[r(t)\y'\p-c(t)\y\p]dt, Ja which makes the base of the so-called variational technique, and the fact that a solution has no zero on a particular interval. As its direct consequence for the interval / = [T,°°), one obtains the equivalence between nonoscillation of equation (1.1) and solvability of Riccati equation (1.6) on [T,°°). The use of this relationship is the basis of the so-called Riccati technique. Its modification, as described later, is the main means of our proofs. Recall that Riccati transformation is widely used also in the theory of second-order linear equations (1.2) and the Riccati equation joined with equation (1.2) through the relation w = reads as w + c(t) + r"1 (t)w2 = 0. (1.5) Chapter 1. Introduction 8 To extend the concept to half-linear equations (1.1), let x be a solution of (1.1) and q the so-called conjugate number to p for which i+i = i. P Q Consider the half-linear Riccati transformation V w(t) = r(t)<$> _ Then a direct differentiation together with substituting from (1.1) gives , (rc(t), r(t)>R(t)>0 (1.8) for t G [^i,^]- Then any solution of (1.7) has a zero in (Ji,^) or it is a multiple of the solution x. The last possibility is excluded if one of the inequalities in (1.8) is strict on a set of positive measure. The concept of principal solution was extended to half-linear equations by Mirzov [105] (and independently by Elbert and Kusano [57], who used a different but equivalent approach). Given a nonoscillatory equation (1.1) and the associated Riccati equation (1.6), it was shown that among all solutions of (1.6) there exists a minimal one w, such that w(t) > w(t) for any other solution w of (1.6) and for large t. The principal solution of (1.1) is then defined by jc = exp|yrr1~*(j)4)~1(w(j))^| i.e., it is the one which is "produced" by the minimal solution w> of (1.6) with respect to the relation w> = r^x1 /x). Note that yp. lfy=yp, then (1.10) is nonoscillatory and it has got a pair of linearly independent nonoscillatory solutions. The principal one hx{t)=t^ (1.11) is known explicitly, and the second one is asymptotically equivalent to the function h2(t)=tRpLlogPt, (1.12) as t —> oo, see [56]. Since the Euler equation in the critical case (O(x,))/ + |*(x)=0 (1.13) is nonoscillatory, it is natural to ask if it can be perturbed with another term so that the perturbed equation remains nonoscillatory. Up to this point, let us extend the the definition of conditionally oscillatory equations. Suppose that equation (1.1) is nonoscillatory and let d(t) be a positive continuous function. The equation (r(t)®(x,))' + [c(t)+nd(t)]®(x)=0 (1.14) is called conditionally oscillatory if there exists a constant jIq > 0 such that (1.14) is oscillatory for /i > jIq and nonoscillatory for /i < jIq. Searching for a perturbation of equation (1.13), so that the perturbed equation is conditionally oscillatory too, leads to the Riemann-Weber (sometimes called Euler-Weber) half-linear equation <*'+(f + ^7,)* \xp and nonoscillatory for \x < \lp, see [58]. The equation with the critical constant wv+ihi&i)™-0 oo. Remark that hi is asymptotically close to the principal solution of (1.16). The idea of looking for perturbations can be repeated and equations (1.10) and (1.15) can be further generalized. For this purpose, denote the iterated logarithm functions by log! t = log?, log** = logjfc_1(logf), £>2, and their products by j L°s/=nio&?- By a generalized Riemann-Weber half-linear equation or generalized Euler-type half-linear equation, we mean the equation {^)y+(\+ny-^ + ^—\^(x)=o, (i.i8) \tp PitPLogjt tPLo&t) where n>2. The equation is conditionally oscillatory with the oscillation constant \i = \ip and naturally extends the Riemann-Weber equation (1.15). In the critical case \x = np, the equation (*(*'))'+(f+E^JW (1.19) is nonoscillatory and its solutions are asymptotically equivalent to the functions hi(t) =tEp±Logh, h2(t)=t'LpllLoglt\ogl+lt, (1.20) for t tending to infinity, as was shown by Elbert and Schneider in [58]. Let us mention that the above-described process of finding further perturbations which remain conditionally oscillatory in the linear case was commented on, for example in [124, p. 133]. If p = 2, then also q = 2, the critical constants yp and \xp become yi = \ki = \, and equation (1.19) reduces to Chapter 1. Introduction 11 Let us now consider general half-linear equation (1.1) and assume that it is nonoscil-latory. Let h(i) be a positive solution of (1.1) such that h'(i) ^ 0 on some interval of the form [To^00) and denote R{t) := r(t)h2(t)\h;(t)\p-2, Git) := r{i)h{i)®{ti{t)). (1.21) Remark that condition (1.9), which is related to the principality of a solution, can be expressed in the form J00 R 1\t)dt = °°. In [47] the authors showed that under the assumptions -^ = oo, liminf|G(*)|>0, R[t) t^oo it is possible to construct a conditionally oscillatory equation seen as a perturbation of (1.1). It is of the form (r(O*(*0)' + c(t) + - 4>(jc)=0 (1.22) hP(t)R(t)(fR-1(s)ds)2 and the critical oscillation constant of this equation is 1 2q From this perspective, if we take the Euler equation with the critical constant (1.13) in place of the general equation (1.1) and use the principal solution of (1.10), i.e., the p-i function h(t) = t '} , in the introduced construction, we arrive at the Riemann-Weber equation (1.15). The proving techniques used in our papers vary and utilize different specialized statements. Here we summarize the most frequent approaches. In studies of equations on the threshold between oscillation and nonoscillation (as, for example, Euler-type equations (1.15) and (1.18)), the classical Riccati technique is not efficient enough. It turned out that the following modification of the technique is suitable in that case. The early version of the modified Riccati transformation worked with two nonoscillatory equations, one of them was supposed to have a solution which is known exactly and the other equation was seen as a perturbation of the first one. To illustrate the idea, suppose that equation (r(t) c(t) for large t, equation (1.1) can be seen as a perturbation of (1.23), according to the rearrangement c(t)-l(G)v-\G\q (1.25) and G is defined by (1.21). The power of this approach is in the fact that the nonlinear function H(v, G) can be under certain assumptions estimated by the function K\G(t)\q-2v2(t), KeR+, which is quadratic in v. The strength of such estimates is dependent on the knowledge of the properties of H(v,G), which has gradually developed. The behaviour of H(v,G) was considered, for example, in [23, 29, 58, 60]. The above described version of the modified Riccati equation was used in many papers by Došlý and his coauthors mainly at the beginning of the century and its derivation is clearly described, for example, in [46]. It is useful especially in the situation when the equation which is being perturbed is the Euler equation (1.13), since one of its solutions is known explicitly. However, it is insufficient to use it with more general equations such as (1.16) and (1.19). In 2010 Došlý and Fišnarová in [23] studied perturbations of both the coefficients r, c and although their modified transformation used again a known solution h(t) of one of the two nonoscillatory equations, their calculation formed a base for a new situation. That is, when h(t) is, instead of being a solution, only a function which is asymptotically close to it (in the formulation, there appears a general function h(t), but the assumptions can ensure the closeness to a solution which is known only asymptotically). The generalized version of the modified Riccati equation, which extends the previous one, was firstly introduced in [60] by Fišnarová and Mařík. Lemma 1.1 ([60], Lem 2.2). Let h(t) be a differentiable function such that h(t) > 0 and h'(t) ý Qfor large ť be Wh = r (0, °°) is called regularly varying (at infinity) of index ů (and we write / G RV(ů)) if lim = Xů for every X > 0. f^oo f{t) If ů = 0, / is called slowly varying. The Representation Theorem (see, for example [115]) says that / G RV(ů) if and only if it can be expressed in the form /W = P«^expj£^^|, where t > a for some a > 0, (p and i/a are measurable functions such that limf^oo (p(t) is finite and positive and limf^oo = 0. Recall that the classification of solutions of half-linear equations in terms of regular variation was deeply described by Řehák, see [115] and references therein. Among other authors who considered half-linear equations in this context, mention, for example, Jaroš, Kusano, Marie, Manojlovic, Tanigawa, see [90, 95, 98, 99]. In our contribution to the topic, we follow up on previous results and continue in finding the asymptotic formulas for solutions of equations at the threshold between oscillation and nonoscillation and their perturbations'. -14- Chapter 2. Asymptotic formulas of nonoscillatory solutions 15 2.1 Nonoscillatory solutions of conditionally oscillatory half-linear equations In this section, we present asymptotic formulas for nonoscillatory solutions of conditionally oscillatory equation (1.22) (r(O*(*0)' + c(t) + hP(t)R(t)(JtR-i(s)ds)2_ 4>(je) = 0 in the cases when p < ^ and p = ^, as was proved in [109]. We also discuss a link to the special case the when the equation which is being perturbed is the Euler equation (1.13) and our asymptotic formulas generalize those of the Riemann-Weber equation (1.15). The first statement deals with the subcritical case p < ^ and provides asymptotic formulas of a pair of linearly independent solutions. Theorem 2.1 ([109], Thm 1). Suppose that (I.I) is nonoscillatory andpossesses a positive solution h{t) such that h'(t) ^ Ofor large t and let f°° dt / —-=oo and liminf|G(OI > 0. J R(t) t^«, 1 v n If p < then conditionally oscillatory equation (1.22) has a pair of solutions given by the asymptotic formula Xi = h(t)( R^1(s)ds\ Li(t), as t ->• oo, where A; are zeros of the quadratic equation |a2-a+# = o and Li(t) are generalized normalized slowly varying functions of the form U(t) = exp |/f R{s)/i-\{T)dT ds} and £,■(*) -> Ofor t -> oo. The second statement focuses on the critical case p = ^ and brings an asymptotic formula for a solution of the equation (r(0*(*0)' + c(t) + 2qhP(t)R(t)(JtR-i(s)ds)2_ 4>(jc)=0. (2.1) Theorem 2.2 ([109], Thm 2). Let the assumptions of the previous theorem be satisfied. Then equation (2.1) has a solution of the form x = h(t)^ftR~1(s)dsSj"' L(t), (2.2) where L{t) is a generalized normalized slowly varying function of the form L(t) = exp jf R{s)rER-l{T)dT ds} ande(t)^0fort^oo. Chapter 2. Asymptotic formulas of nonoscillatory solutions 16 The proofs utilize the modified Riccati technique in combination with finding a fixed point of a certain integral operator with the use of the Schauder-Tychonoff fixed point theorem. Consider the application of the theorems to the case when the equation which is being perturbed is the Euler equation with the critical constant (1.13). It means that p-i r(t) = l,c(t) = Jpt p and h(t) = t '} . Then conditionally oscillatory equation (1.22) reduces to Riemann-Weber equation (1.15). Indeed, for t —> °° we have R(t) = rh2\h'\P-2= {^y-J t, fR~l{s)ds~ (y-^JP loS' and hence hP(t)R(t)(fR-l(s)ds)2 tP\og2t Observe that for p, = ^ the constant in the last expression is p-iy-2 i P J 2q 1 Asymptotic formulas for fx < ^ (and equivalently ji < jip in (1.15)) then are 2h,2 z± I I p \P-2 \ P z±_ 2Kl *i,2~t p ( (-- ) log? ) = Const-t p log? p as?— which coincides with the results of [108]. There the asymptotic formulas were found in the form p-i x\,i{t)^t p (log?)Vl>2 as?-^oo? 1 ( —\\P where = p- f J and are roots of A2 4ß = 0. ■p It is not hard to check that —^ = Vi^, respectively, and so both the ways lead to the same result in a qualitative sense. The second theorem can be applied to the Riemann-Weber equation in the critical case (1.16), i.e., to the equation 4>(je) = 0. tP ' tP\og2t_ formula asym as t tends to infinity (see (1.17), as was proved by Elbert and Schneider in [58]). p-i i The formula (2.2) then reduces to the formula asymptotically close to h\(t) = t p Xogp t Chapter 2. Asymptotic formulas of nonoscillatory solutions 17 2.2 Subcritical half-linear Euler-type equation The paper [110] is devoted to finding a nonscillatory solution of generalized Euler-type equation (1.18) with n terms in the sum (1.18), i.e., W))' + [\ + E + —tL-r) = 0 in the subcritical case for /i G (0,nP). The found asymptotic formula can be seen as an extension of that type of result for nonoscillatory solutions of Riemann-Weber equation (1.15), which was proved in [108]. The main statement reads as follows. Theorem 2.3 ([110], Thm 3.1). Equation (1.18) with jl G (0, jlp) has a solution of the form p-i I 21 x(t) =t p Log^log/ f(l+o(l)) as (2.3) where X = \ + \sj\-fp. We showed that the solution (2.3) of equation (1.18) is a regularly varying function of index and the function (1 + o(l)) in its formula is a slowly varying function. The proof is based on closeness of the modified Riccati equation (1.27) for equation (1.18) v'«) + I 77^27 + + ^H(v(t),2nP)=0 pit Log jt thog^t t and the so-called approximate modified Riccati equation u(t) + £ -^V + —^2- + IT-At) = 0. (2.4) PitLogjt tLog2nt \)iPt It is possible to find the solution of (2.4), and it serves in constructing a suitable set, on which a certain integral operator is considered, and such that the Schauder-Tychonoff theorem can be used. The closeness of both the equations is shown with the use of estimates for the function H(v, G) from [23]. Chapter 3 Oscillation and nonoscillation criteria One of the most important tasks of the qualitative theory of differential equations is to provide the oscillation and nonoscillation criteria. With their use, one can decide, based on the form of the coefficient functions, whether an equation is oscillatory or nonoscillatory. In the first part of this chapter, we focus on extensions of Hille-Nehari type criteria for conditionally oscillatory half-linear equations. The latter part presents a generalization of an integral comparison theorem of Hille-Wintner type. Perturbations of conditionally oscillatory half-linear equations are considered. 3.1 Hille-Nehari type criteria Throughout the section, we suppose that equation (1.1) is nonoscillatory and we study its perturbation in the form (r(t)(, (B) where h is a solution of equation (1.1) (or a function which is asymptotically close to a nonoscillatory solution of (1.1)). Recall that the function R(t) is defined by (1.21). 3.1.1 Euler-type equation Let us summarize the results we will follow up on. The perturbed half-linear Euler equation (% + d(tj)*(x)=0 (3.2) Chapter 3. Oscillation and nono s dilation criteria 19 was considered in [35], where it was shown, under the notation d(s)sp~1ds, that equation (3.2) is nonoscillatory if limsup£(ř) < liminf£(ř) > —3/ip and oscillatory if liminfE(i) > /ip. The same couple of nonoscillation and oscillation criteria with the integral expression E(t) = Íd(s)sp-1 log2 sds Xogt Jt was then proved by Došlý and Řezníčková in [40]. Let us point out that the relation p-l p-l 2 between E{t) and types (A) and (B) is given by taking h{t) =t p and h{t) =t p Xogp t, respectively, which are (almost) solutions of Euler equation (1.13), see (1.11) and (1.12). Now let us turn our attention to the perturbed Riemann-Weber half-linear equation with critical coefficients +{7i + -J^t:+Ji*)) = o- (3-3) The criteria in terms of the integral expression /oo d{s)sp~llogsds, p-i i that complies with (A) taking h(t) =t p Xogp t, which is close to the principal solution of (1.16), were proved in [19] (the nonoscillation criterion) and [46] (the oscillation criterion). p-l I 2 The case which corresponds to type (B) and to the function h(t) =t p Xogp tXogp (logř), which is asymptotically close to the second solution of Riemann-Weber equation (1.16), see (1.17), remained open. Finally, we come to the perturbations of general half-linear Euler-type equation with critical coefficients (1.19), i.e., to the equation LRW[x] := (<*>(*'))'+ + t T^T-A^n *W = °- (3-4) Došlý in [20], with the use of the function h(t) = t p Log£t (i.e., h\(t) from (1.20)), proved the following statement. Theorem 3.1 ([20], Thm3.3). Suppose that the integral J°° d(t)tp~l hogntdt is convergent. Chapter 3. Oscillation and nono s dilation criteria 20 (i) ¥ />oo limsuplogM+1f / d(s)sp~l Lognsds < jlp, t—> Jt /■oo liminflog , ,t / d(s)sp^1 Log sds > — 3/ip, then (3.4) is nonosdilatory. (ii) Suppose that there exists a constant y > ^^jf^? sucn that d(t)tp log31 > J for large t and poo liminflog , ,t / d(s)sp~iLog sds > jlp. Then (3.4) is oscillatory. If n = 1, then (3.4) reduces to (3.3) and the criteria from Theorem 3.1 reduce to those obtained in [19, 46]. In our paper [65] we continued further with complementing Theorem 3.1 (and the corresponding results of [19, 46] in the case n = 1). We utilized the second function p-i I 2 h/i = t p LogntlogP+lt from (1.20) and found a related couple of criteria for equation (3.4) formulated in terms of the expression 1 ŕ ? -- / d{s)sp Logns\ogn+lsds n+ll J log which corresponds to type (B). The results of [65] are presented in the following statements. First, we present the nonoscillation criterion. In comparison to previously mentioned nonoscillation criteria, instead of the constant \xv on the right hand side of the inequality we have a class of constants, which enlarges the scope of the result. Theorem 3.2 ([65], Thm 3.1). if limsup- / d(s)sp^i Lognslogl+lsds <2idp(-a+ V2a), (3.5) f^oo logM+11 Jt liminf-- / d(s)sp Log slogz,, sds > 2/ip(-a - v2a) (3.6) t^oo l0gn+ltJT for some a > 0, then equation (3.4) is nonoscillatory. Recall that if a = \, then 2\ip{-a + \/2a) = /ip, 2/j.p(-a-Via) =-3/j.p and the constants from (3.5) and (3.6) in Theorem 3.2 reduce to the constants in Theorem 3.1, part (i). The following statement is the oscillatory criterion which complements Theorem 3.2. Chapter 3. Oscillation and nono s dilation criteria 21 Theorem 3.3 ([65], Thm 3.3). Suppose that there exists a constant y such that V i'Log^ 3(p-l)2 Tor Zarge /. 7/" liminf-- / d(s)sp Log„slog„ , ,sds > }lp, ř^°° logn+iř7r then (3.4) oscillatory. In the proofs we utilized general statements from [26] and [61]. There we employed the expansion of the term h(t)LRw [h] (t) with the use of the function h(t) =t p Log „ t log^+11 and of the term h(t)LRw[h](t) with the function h(t) = t p Log^ř. 3.1.2 General conditionally oscillatory equation Perturbations of general conditionally oscillatory equation with the critical coefficient (2.1) were studied firstly by Došlý and Ůnal in [47]. One of its results (Theorem 5) provides a nonoscillatory Hille-Nehari type criterion for the equation c(t) H--—t-?+g(t) 2qhP(t)R(t)(fR-1(s)ds)2 4>(je)=0 (3.7) (r(0*(V))' + and the integral expression log (^JlR-1(s)ds^j jf" g(s)h?(s) JSR-\x)dxds is employed. Note that the result is of type (A). The crucial role in the proof of this criterion plays the fact that the asymptotic formulas for solutions of (2.1) are known. It was shown in [21,47] (see also [109]) that equation (2.1) has a pair of linearly independent solutions that are asymptotically close (as t —> °°) to the functions xi(t)=h(t)^j R~1(s)dsj , (3.8) x2(t)=h(t) (^JlR-1(s)dsy' logp (^JlR-1(s)ds^j . Our contribution to the topic was presented in the paper [66]. We improved the above-mentioned nonoscillation criterion for (3.7), formulated a relevant oscillation criterion for (3.7) and found a perturbation g(t) in (3.7) such that the equation becomes conditionally oscillatory. We also formulated a version of nonoscillation and oscillation Hille-Nehari type criteria for equation (3.7) with the use of the asymptotic formula for the second solution of equation (2.1) (i.e., of type (B)). Chapter 3. Oscillation and nono s dilation criteria 22 The first theorem of [66] improves [47, Theorem 5]. The main difference is that we get by the condition limlog2 (jR-i(s)ds)R(t)G'(t) =0 considered in [47] and that the inequalities hold for a set of constants. Theorem 3.4 ([66], Thm 3.1). Suppose that h is a positive solution of (1.1) such that h'(t) t^O for large t, /oo R-i(t)dt = °° and liminf\G(t)| > 0, (3.9) hold and the integral J°°g(t)hp(t) f R~x{s) dsdt converges. If limsuplog^y R-\s)ds^J jf g(s)hP(s) J RTl(x)dxds < ^(-« + V2a), liminflog (^JlR-\s)ds^j jf g(s)hp(s) J* RTl(x)dxds > ^(-a-V2a) for some a > 0, then (3.7) is nonoscillatory. Note that the choice a = \ reduces the right hand sides of the inequalities to the constants ^ and —j^, respectively, which appear in [47, Theorem 5]. Let us briefly describe the main characteristics of the other statements from [66] without all the details. In all of them, condition (3.9) is assumed. In [66, Thm 3.2], an oscillatory counterpart of Theorem 3.4 is presented. It summarizes the conditions, which together with the inequality liminflog (^f R~l (s) ds^j jf" g(s)h?(s) J''R~l{x)dxds > ensure that equation (3.7) is oscillatory. Note that the above mentioned results are of type (A). Next results of [66] are oscillation and nonoscillation criteria of type (B), formulated in terms of the integral expression Finally, it is shown that for the perturbation g(t) of the form g(t)-=-;-;-=-n—t-;-, <5 G IR, (3.10) W hP(t)R(t)(fR-1(s)ds)2logl(fR-1(s)ds) equation (3.7) becomes under certain conditions conditionally oscillatory. The precise formulation is presented in the following theorem. Chapter 3. Oscillation and nono s dilation criteria 23 Theorem 3.5 ([66], Thm 3.3). Suppose that h is a positive solution of (1.1) such that h'(t) 7^ Ofor large t and (3.9) holds and consider the equation (r(t)))> + j^ and there exists a constant j5 such that R{t)\og2{jsR-\x)dx) ~ G2^ holds for large t, then (3.11) is oscillatory. The proofs of the statements from the paper [66] refer to oscillation and nonoscillation criteria [61, Theorem 3.1 and 3.2], [26, Theorem 1 and 2]. The choice of the used helping function was x\ and x2 from (3.8) and h= (J R )p logp (J R ), according to the conjecture that this function is asymptotically close to one of the solutions of (3.11) with 8 = ± 3.2 Hille-Wintner type integral comparison theorems Consider equation (1.1) together with the equation L[x] := (r(t)(, the Hille-Wintner theorem says that if (3.12) is nonoscillatory and /oo />oo c(s)ds< J C(s)ds for larger, then (1.1) is nonoscillatory too. If J°°r1 q(t)dt < °°, denote k(t) := //"r1 q(s)ds and assume that c(t) > 0, C(t) > 0 for large t. Then if (3.12) is nonoscillatory and /OO /"OO c(s)kp(s) ds < C(s)kp(s) ds <<>° for large t, equation (1.1) is nonosillatory too (for both statements see [39, Section 2.3.1]). Our work in this field utilizes adopting the perturbation approach, which leads to results applicable to equations on the threshold between oscillation and nonoscillation. Together with (1.1) and (3.12), consider the equation of the same form (1.23) which is supposed to be nonoscillatory and denote its left hand side by We look at equations (1.1) and (3.12) as if they were the perturbations of nonoscillatory equation (1.23). The following theorem is the main result of the paper [37] and reads as follows. Chapter 3. Oscillation and nono s dilation criteria 24 Theorem 3.6 ([37], Thm 1). Let J°° rl~q(t)dt = °°. Suppose that equation (1.23) is nonosdilatory and possesses a positive principal solution h such that there exists a finite limit lim G(t) > 0 and /oo R-1(t)dt = °°. Further suppose that 0 < Jf°° C(s) ds < °° and /OO POO (c(s)-c(s))hp(s)ds < J (C(s)-c(s))hp(s)ds <°°, all for large t. If equation (3.12) is nonoscillatory, then (1.1) is also nonos dilatory. A direct application of the theorem leads to the corollary for the case when the equation which is being perturbed (1.23) is the Euler equation with the critical coefficient (1.13). We know its principal solution explicitly. Unfortunately, for Riemann-Weber equation (1.16) and Euler-type equation with n terms in the sum with the critical coefficients (1.19), only asymptotic formulas of the principal solutions are known. Hence, the theorem cannot be applied. The question, if the principal solution in Theorem 3.6 can be replaced by a function close to it, remained open for several years. Thanks to the fact that the modified Riccati technique has been deeper developed, the answer could have been found. In [111] we proved the following results. Theorem 3.7 ([111], Thm 1). Suppose that there exists a positive continuously differen-tiable function such that h'(t) ^ Ofor large t and the following conditions holds: /oo R-\s)ds = °°, /oo h(s)L[h](s)ds 0 and limsup\G(t)\< °°) or lim\G(t)\ = °°, f^oo f^oo all for large t. Let the inequality ) poo poo h(s)L[h](s)ds< J (c(s)-c(s))hp(s)ds< J (C(s) - c(s))hp(s)ds < °o be satisfied. Then if equation (3.12) is nonoscillatory, equation (1.1) is nonoscillatory too. As an immediate consequence of the previous theorem, we see that under the assumptions of Theorem 3.7 oscillation of equation (1.1) implies that of (3.12). Next comes the consequence of Theorem 3.7 for the case where the nonoscillatory equation (1.23), which is being perturbed, is the Euler-type equation with the oscillation constant (1.19). Denote its left hand side by LRW[x] := (*(*'))'+ + t tJ^x) (3-13) Chapter 3. Oscillation and nono s dilation criteria 25 p-i Recall that Došlý in [20] showed that for h(t) = t p Log%t and the operator defined in (3.13) we have tlog^t I 3(p- l)z Corollary 3.1 ([111], Cor 2). Suppose that the condition as t —> oo. (3.14) ř ^ Log£r >0 holds for large t and take cit) + 1 Vp tP ' pltP\jog)tt If the inequality /oo poo poo fds< J (c-c)sp-1Logn(s)ds< J (C-c)sp-1 Logn(s)ds to and 1. We suppose that the coefficients of the equation satisfy the conditions r e C([to,°°),'R+), b e C1([to,°°),M.q ), c e C([?o,°°),Ro ), c is not identically equal to zero on any neighborhood of infinity and b(t) < 1. Concerning the deviating arguments, we assume that x, o e C1 ([to, °°), IR), limf^oo x(t) = °°, limf^oo 0, x(t) < t, o(t) < t. We also suppose that /oo rl-«(t)dt = oo, where q denotes the conjugate number of p, i.e., q = j^j. The above setting and conditions are supposed to hold throughout the whole chapter. Clearly, equation (4.1) can be seen as a generalization of equation (1.1) to which it is reduced for x{i) = t and b(i) = 0. However, oscillation properties of both the equations significantly differ and equation (4.1) evinces more complex and complicated qualitative behavior. By a solution of (4.1), one understands a differentiable function x{i) (x(t) ^ 0), such that r(t)<&(z!(t)) is differentiable and (4.1) holds for large t. Equation (4.1) is called oscillatory if it does not have a solution which is eventually positive or negative. Contrary to equation (1.1), oscillatory and nonoscillatory solutions can coexist and hence the existence of one oscillatory or nonoscillatory solution does not ensure that equation (4.1) is oscillatory or nonoscillatory, respectively. One of the possible ways to prove oscillation criteria for (4.1) is via the Riccati technique and our contribution to the topic was to present some criteria which can be obtained by -26- Chapter 4. Neutral half-linear equations 27 its modification (cf. Chapter 1.2). The (modified) Riccati method here allows to work only with Riccati type inequality, in contrast to Riccati equations (1.6) and (1.27) reated to equation (1.1). In more detail, suppose that equation (4.1) has an eventually positive solution x{i). Take At) w(t) = r(t) z(t(0) A direct differentiation yields w{t)= *(z(t(0)--(p"1)rwtw |z(t(0)|^ which, with the use of the inequality (rWO(z'W))/<-c(^[z(TW)(l-M^))] proved in [67, Lemma 1], gives w(t) < -cm (i - b(xit))) ~(p- \y-^t)x\t)z^^\W{t) i< Assuming that there exists a positive function f(t) such that z'(x{t)) z'(t) >f(t)>0 (4.2) we obtain the Riccati type inequality of the form w\t) < -cm (i - *(t(0)) -(p- \y-\t)x\t)f{t)\w{t) i«. To continue with the modification of the Riccati technique, let h(i) be a positive differentiable function and denote G{t) = r{t)h{x{t))<$> h'(x(t)) fit) (4.3) Using the modified Riccati transformation v{t)=hp{x(t))w(t)-G(t) provides (see [67, Lemma 3]) the so-called modified Riccati inequality i/(0 + C(t) + (p- iy-«(t)x'(t)h-«(x(t))f(t)H(v(t),G(t)) < 0, where C(t)=h(x(t)) (4.4) and H(v, G) is the function defined by (1.25). Chapter 4. Neutral half-linear equations 28 4.1 Oscillation criteria The first theorem of [67] can be seen as a result of the same character as, for example, [70, Theorem 5] or [120, Theorem 2.1]. In both the mentioned papers, a variant of Riccati transformation is used as the proving technique, whereas we use the modification of the Riccati technique as described in the previous section. The statement reads as follows. Theorem 4.1 ([67], Thm 1). Let f be a positive function, G, H be defined by (4.3), (1.25), respectively, h be a positive continuously differentiable function such that h' ^ Ofor large t and C(t) > Ofor large t. Moreover, let either limsup|G(f)|°° and *(/(0M0 ■dt it)h2(T(t))\h'()>W_2Ly (4.8) This leads to the following consequence. Denote R(t) = r(t)^h2(x(t))\h\x(tW-2f-p(^ (4.9) Under the assumptions of the paper, according to (4.8), we can take f(t) = <£> 1 (r(r(f))) and the functions G, C and R (see (4.3), (4.4) and (4.9), respectively) get the following form: G1(t) = r(T(t))h(T(t))(T(t))), Ci(0 = M<0)[(K<0WW0))), + c(0*(W)))*(i-W0))]> l TO' In this particular case of the function /, we can formulate a version of Theorem 4.1 as follows. Ri{t) = r{T{t))—h2{T{t))\ti{mr2- Chapter 4. Neutral half-linear equations 29 Corollary 4.1 ([67], Cor 1). Let h be a positive continuously differentiable function such that hi ^ Ofor large t and C\ (t) > Ofor large t. Moreover, let either f°° x'(t) limsup|Gi(*)| < °° and / -, . . . . NN dt = °° (4.10) f^ooPI Wl J r^-l{x{t))m{x{t)) or /oo R7l(t)dt = oo. Ci(t)dt = oo, (4.11) then equation (4.1) is oscillatory. The second result of [67] presents a Hille-Nehari type criterion which can be again applied to the same situation as in the preceding Corollary 4.1. Theorem 4.2 ([67], Thm 2). Let h be a positive continuously differentiable function such that h' ^ Ofor large t and f be a positive function such that C(t) > Ofor large t, /oo R 1 (t) dt = oo, liminf Git) > 0 and limsupG(7) < oo or limG(*)=°o. (4.12) f^oo f^oo Suppose that J°°C(t)dt < °°. If /t poo 1 R-1(s)ds C(s)ds>—, (4.13) Jt 2q then equation (4.1) is either oscillatory or in every neighborhood of°° there exists t* such that zt(f*^yti^ < 1 for all solutions of (4.1). 4.2 Application to neutral Euler-type equations Consider the Euler-type equation (®(x(t)+box(o(t)))'y + £(x(Xt))=0, (4.14) where A G (0,1), a(t) < f,limf_>oo(l-b0) l(Xt)^ + y(1 — bo)Xp 1 implies (4.11). Thus, by Corol- lary 4.1, equation (4.14) is oscillatory if 7> p-l 1 XP-1<5>(1 - bQ) (4.17) This corresponds to the result known in the case bo = 0 (equations with delay, see [63]) and also to the case bo = 0 and X = 1 (ordinary equations). Grace et al. showed in [70] that under some additional assumptions, condition (4.8) can be strengthened. Similarly as in [125], they considered the sequence So(p):=l, gn+l(p):=eP^P\ n = 0,1,2,..., (4.18) where p is a positive constant. For p G (0, the sequence is increasing, bounded above and limf^oo^w(p) = g(p) e [l,e], where g(p) is a real root of the equation With the use of this sequence and the notation g(t):=t\, where t\ is large enough, Grace et al. proved in [70, Lemma 4] that if x(t) is strictly increasing, equation (4.1) has a positive solution x(t) on [z*o,°°) and the condition £{s)<&{Ě{s))ds>p (4.20) holds for some p > 0 and t large enough, then Z'(TW) (gn(P)r(t) z'{t) - V r(x(t)) for every n and t large enough, where gn(p) is defined by (4.18). Chapter 4. Neutral half-linear equations 31 Thanks to the preceding estimate, condition (4.17) can be even strengthened for a class of equations (4.14), which satisfy (4.20) with p G (0, ^). By a direct computation (or see [70]), one can show that for the considered Euler-type equation (4.14) we have p = (i-^-1^-1^i + -lTA^-1(i-fco)/,-1y)/' 'log^iy). If p G (0, i), we can use in place of the positive function f(t) from (4.2) the function ^ 1 i^r{-!{t)) )' wnere s{p) is defined by (4.19). Then functions G, C, R according to their definitions (4.3), (4.4) and (4.9), respectively, become G2(t) = r(T(/))Ä(T(0 WMOWSÍP))-1 C2(t)=h(T(t)) *(P) *2(0 = r(TW)^2(TW)|^(T(0)|^2U(p))-1. These in our setting for equation (4.14) and again with h(t) = t~ read as Gi{t)=(^]P \g{p))-\ Ri(t) n_\\P-2 P A (siPT't and C2(t)=f (g(p))-l + y {^y- holds, which is the ordinary version of condition (4.17). The constant yp = {^y^j is the critical constant between oscillation and nonoscillation of (4.14), and it is natural to study perturbations of the Euler-type equation with this critical constant and to find critical constants in the added terms. In the case of the delayed and neutral equations, there is not such a boundary between oscillation and nonoscillation of (4.14). However, based on the results known from the ordinary case, let us study the neutral version of the Euler-Weber type equation (4K*(0+M(jc(A(ř))=0. (4.21) Chapter 4. Neutral half-linear equations 32 For equation (4.21) we use Theorem 4.2 with f(t) = <£> 1 (r(^))) • Similarly p-1 as previously, we take h(t) = t p , for which (4.15) holds, and with the use of the relevant coefficient function c(t) in (4.16) we have ci(0 p tlogzt ^(l-b0)Xp-1 and C\ (s) ds t Jt ,p-2 H(1 - bQ)XP-1 s log2s logt Since 1 = (^~r) \> condition (4.13) becomes which, by Theorem 4.2, implies that equation (4.21) is oscillatory if P> 1 24>(1 - b0)XP-1 V P p-l (4.22) Note that in the ordinary case X = 1 and bo = 0, the constant from (4.22) is critical, which means that (4.21) is oscillatory if and only if /i > j ( -— j Finally, let us consider the perturbation of the Euler-Weber equation (4.21) with \x = \xv. p-i i Take h(t) = t p Xogp t and consider equation (4.1) with r(t) = 1, x{t) = Xt, b(t) = 0. Observe that 1 1 P — 1 -I I H{t) =--1 p logp t 1 + (p-l)logf. By direct computation, we see that Gi(0 = *i(0 = p-l p-l p-i log(A*)(l + o(l)), p-2 flog(Af)(l+o(l)) as t —> oo. With the use of the power expansion formula (l+x)s = l+sx+^——Lxl + s(s — 1) 9 s(s — l)(s — 2) o , a. v ;-2 1 v -^--x3+o(x3) as x^O one can show that for c(t) Tp +l( P p-i + ■ P XP-ltP 2\p-\) XP-hP\og2{Xt) ^ log2 (A?) log2 (log (A?)) (4.23) we have ci(0 \iXP -1 flog(Af)log2(log(Af)) (l + o(l)) asf-^oo. Chapter 4. Neutral half-linear equations Because 2-p JR^\t)dt~(?-pl log(log(AO) and Ci(t)dt< ^P log(log(Af)): as t —> °°, condition (4.13) becomes 2-p 1 = p-1 A?, b(t) = 0 and with c(t) given by (4.23) is oscillatory if Since - = £—, the considered perturbed Euler-Weber equation (4.1) with r(t) = 1, x{i) l/p-iy-1 l 2 \ p J Xp-1' One can see that the constant is reduced to jlp in the undelayed case X = 1. Chapter 5 Numerical approach Within the numerical approach to finding approximate solutions of differential equations, we focus on a semi-analytical method based on the differential transformation. Contrary to other numerical methods, where the result is a set of points and approximate function values, we arrive at an approximate polynomial. In [114] we proposed an algorithm using the differential transformation, which is convenient for finding numerical solutions to initial value problems for functional differential equations. Our focus was on retarded equations with delays which in general are functions of the independent variable. The main idea of the algorithm is that a delayed differential equation is turned into an ordinary differential equation using the method of steps, and subsequently, the ordinary differential equation is transformed into a recurrence relation in one variable using the differential transformation. The approximate numerical solution has the form of a Taylor polynomial whose coefficients are determined by solving the recurrence relation. In the paper [112], the algorithm is used for finding approximate numerical solutions to second-order half-linear Euler equation with and without delay, and recurrence equations for Taylor series coefficients of solutions to initial value problems are derived. Here we recall the basics of the differential method in such a depth, which is necessary for understanding how the application to the Euler equations is made, and we present the process, how the approximate solutions for half-linear Euler equations in the form of Taylor's polynomial can be found. 5.1 Differential transformation method Differential transformation of a real function u{i) at a point to e M. is where U(k) [to], i.e., the k—th component of the differential transformation of the function u{i) at to, k e Nq, is defined as @{u(t)}[to] = {U(k)[to]}k=t (5.1) U(k)[t0] l_ \dku(t) T\ dtk J t=t0 (5.2) -34- Chapter 5. Numerical approach 35 provided that the original function u(t) is analytic in a neighbourhood of to. Inverse differential transformation of {U(k) [to]}^=0 is defined as u(t) = {U(k)[to]}U\[to] = £ U(k)[t0](t-to)k. (5.3) k=0 In applications, the function u(t) is expressed by a finite sum n u(t) = Y,U(k)[t0}(t-to) k=0 k Now we present the formulas for the differential transformation of such functions, which we need for transforming the half-linear Euler equation. They can be found, for example, in [114]: Lemma5.1. Assume that F (k)[to\ andU(k)[to] are differential transformations of functions f(t) and u(t), respectively. If/it) = then F{k) M = ^±|^V(* + n) [»„]. (5.4) ///(,) = ,V £ R, ,Hen F(kM = QVfratt , suCH ,Ha, |, - *| < W, (5.5) J =-—-= —tti (r)k being the Pochhammer symbol. K I /CI /CI k If At) =g(t)h(t),thenF(k)[to] = Y,G(l)[to\H(k-l)[to\. (5.6) 1=0 Differential transformation of components containing nonlinear terms can be obtained using the algorithm described in [113]: Lemma 5.2. Let g and f be real functions analytic near to and g(to) respectively, and let h be the composition h(t) = (fog)(t) = f(g(t)). Denote @{g(t)}[t0] = {G(k)}^=0, ®{f(t)}\g(*o)] = {F(k)}k=o and ®{(f°8)(t)}[to\ = {H(k)}k=o the differential transfor-motions of functions g, f and h at to, g(to) and to respectively. Then the numbers H(k) in the sequence {H(k)}^=0 satisfy the relations H(0) = F(0) and k H(k) = £ F(l)-BKl (G(l),..., G(k- I + 1)) jbrk>\, (5.7) l=i where Bk,i(xiT.. are the partial ordinary Bell polynomials. Here, according to [113], the partial ordinary Bell polynomials B^xi,... I = 1,2,..., k > I, satisfy the recurrence relation k-l+i -j Bk,i (*i,•••, Xk-i+i) = -[-XiBk-i,i-1 (*i, • • •, Xk-i-i+2), Chapter 5. Numerical approach 36 where Bqq = 1 and B^q = 0 for k > I. Finally, we need the following result which is proved in [114]. If ^{f(t)}[to\ = {F(k)[to]}^=0 is the differential transformation of the function f(t) at to. Then the components F(k)[ti] of the differential transformation @{f(t)}[h] = {F (k)[ti]}%=0 of f(t) at t\ > to may be expressed as F(k)[ti] = JL( .J)(h-to)jF(k+j)[to], k>0. (5.8) j=o V J J 5.2 Application of the differential transformation to half-linear Euler equation In the paper [112], the differential transformation method is applied to half-linear Euler equation in the form ((^(0)°), + ^r^(0=o, (5.9) as well as to the second-order half-linear Euler equation with a proportional delay ((x(t))ay+-^Ixa(xt) = o, where a > 0 is a quotient of two odd positive numbers and X e (0,1). Note that a = p — 1 and equation (5.9) is a special case of equation (1.10). Consider the initial value problem for half-linear Euler equation ((■r/)a)/ + ^TT^a = 0, x(to)=a, x\to)=b. (5.10) To apply the differential transformation of (5.10) at to, we use Lemma 5.1 and successively obtain: @{x(t)}{5= {X(k)}7=o, X(0)(5=a, X(l)(5^b, ®{x'(t)}{5^ {(k+\)X(k+\)Tk=Q, ^{(*;«)a}(5=^ =:{#i(*)}£=i > Hi(0) = (ao)(X(l))a, ®{((At))a)'} =4) i(k+ m(k+ l)>r=o: ^{W0)°}(5= jl (")(X(0))«-^(X(1),X(2),...)| =: {H2{k)Tk=l #2(0) = (")(X(0))( Chapter 5. Numerical approach 37 Hence equation (5.10) transformed at t = to, with use of the rule for transforming product (5.6), reads as k (k+l)Hl(k+l)+rY,Fi(l)H2(k-l)=0, k>0. (5.11) 1=0 This is a recurrence relation, to which it is possible to substitute the initial conditions and then compute successively further members of the sequence X. In the end, the inverse transformation (5.3) is applied. Example 5.1. As a concrete example, take Euler equation (5.10) with a = 3. We know that if 7 = 73 = (|) , then t3/4 is a solution of the initial value problem ,4 ((x')3Y + ^-x3=0, jc(1) = 1, J{1) = \ (5-12) Let us illustrate the process of finding the Taylor series of the solution on this problem (5.12). Transformed equation (5.11) expands for (5.12) with a = p—l=3,to = l,J= (f) to jfc+i 0 = (*+ 1) t (')J(2X(2),...) +rE (?) E Q wr'Wm■ ■ ■).+r(-t4) g)(*(o))3 The initial conditions imply X(0) = 1 and X(l) = |. Substituting for k = 0,1,2 provides the results 3 5 45 X(2) = --, X(3) = —, X(4) 32' v ' 128' v 7 2048 The precise calculations can be found in [112]. We see, that using the differential transformation algorithm we get the Taylor series coefficients of the exact solution. Indeed, Taylor series expansion of t3^4 at to = 1 is 3 3 5 45 i + jC-D-jjC-D' + ijgC-D'-joigC-^ + tXC-i)5), l'-H 1- lit E Ii then Xt lies in For t E I\ = [to, t\] we denote the solution of initial value problem (5.13)+(5.14) as x\ (t) and its differential transformation at to by X\ (k) [to], k>0. Since Xt for t El\ falls into Iq, we substitute fovx(Xt) the initial function (5.14) and rewrite equation (5.13) into the form (W(t))ay + -^I(a + b(Xt-to))a = 0, x(to)=a, x'(t0)=b. (5.15) Because a + b(Xt-to) = a + bt0(X - I) + bX(t - to), we have 3l{a + b(Xt-to)}[to] = {a + bto(X-l),bX,0,0,...}, 3?{(a + b(Xt-to))a}(k)[to] i5=t ^(a + bto(X-\))a-lBKl(bX,Q,Q,...) (a + bto(X-l))a-k(bX)k,k> 1, k $l{(a + b(Xt-to))a}(0)[to] = (a + bto(X-l))a. Equation (5.15) transformed at to with the use of the rule for transforming the product (5.6) reads as ^+0l(")(Zi(0M)a^+i,z(2X1(2)[^,3X1(3)M...) (5.16) + rt ("V l) (to)-"-1-1 z) (a + bto(X - l))"-*+'(*A)*-' = 0 k I 1=0 and the initial conditions transform, according to (5.2), to X1(0)[t0]=a, Xl(\)[to]=b. Substituting for k = 0,1,... into (5.16) provides recurrent equations, from which one can successively calculate X\ (k) [to] for k > 2. Resulting solution on interval I\ then is oo Xl(t) = Y,Xi(k)[to](t-t0)k. k=0 Chapter 5. Numerical approach 39 In the applications and programming, this series (and every other) will be truncated. The solution x\ will be just approximate. Notice that with a different (general) initial function 0, equation (5.16) would be of the form: + 1) (7) ^Xl^^^""'^+1,/(2XX(2)[?0],3XX(3)[^o] • • -) Now let us move to the second step, take t G h = [h,t2], denote X2(t) approximate solution on I2 and X2(k)[t\], k > 0, its differential transformation with the center at t\. Because Xt for t 1, 1=1 $}{xl(Xt)}(0)[h]=Xl(0)[t0]=a=:G(0)lh]. Furthermore, again according to the way how to transform a compound function (5.7), 0{(jci(AOH(*)[*i] = E (D (G(0)h])°-/^./(G(l)h],G(2)h],...),*>0. 1=0 \L / Equation (5.17) transformed at t\ reads as 0=^+1)1 ^(X2(l)[/i])0-/^+i>/(2X2(2)[/i],3X2(3)[ft]...) (5.18) 1=0 V 1 / j=oVJ/ and the initial conditions are (according to (5.8)) 00 X2(0)[h]=Xi(0)[h] = £Xi(*)[*0](*i -t0)k k=0 OO X2(m]=Xl(\)[tl] = Y,(k+l)Xl(k+l)[t0}(ti-to)k. k=0 Chapter 5. Numerical approach 40 The approximate solution for t El2 then is oo *2(*) = £x2(*)[*i](*-*i)*. k=0 Further steps for t E Iu i > 3 employ again recurrent equation (5.18), only with shifted indexes (e.g., X2 becomes X, and t\ becomes The process of the i-th step follows the path of the second step. The paper [112] presents numerical results for a concrete example of the initial value problem (5.13)+(5.14) together with their comparison with results obtained by Matlab function ddesd. In particular, the following problem is considered ((^))3)/+(^)4^W0.8?))3 = 0, *(1) = 1, x'(l) = ^ (5.19) with the initial function 0(0 = l + |(f- 1), te(0,l]. (5.20) The first step of the algorithm takes place on the interval [to,ti] = [1, ^] and the second step on the interval [t\,t2] = (t>ecause U = |r)- The results are illustrated in the figure. There we see, how the approximate solutions of the delayed problem (5.19)+(5.20) tend to rise more at the end of the interval [^0,^2] compared to the original undelayed solution of the initial value problem (5.12). We can observe, and it is discussed in more depth above the table of numerical results in [112], that our algorithm provides an approximate solution that is very close to the solution calculated by the built-in Matlab function ddesd designed for solving delay differential equations. Conclusion We conclude the thesis with some remarks presenting further possible research directions and open problems and with a brief summary. Remark 5.1. Further possible work in a search for asymptotic formulas of nonoscillatory solutions is straightforward and concerns the generalized Euler-type equation in the sub-critical case (2.2). The formula for its second nonoscillatory solution linearly independent to (2.3) is still to be found. Another potential direction would be turning the attention to formulas for oscillatory solutions. Some of the formulas for oscillatory solutions of (1.22) in the supercritical case, where proposed by Došlý in [21], but as far as we know, this area remains unexplored. Remark 5.2. Based on the results of Section 3.2 for perturbations of general conditionally equations and their comparison with those for the perturbed Euler-type equation in Section 3.1, we suppose that we can study perturbations of equation (3.11) with 8 = ^ and find a perturbation such that the obtained perturbed equation is conditionally oscillatory. More generally, we conjecture that the equation with an arbitrary number of iterated logarithmic terms ((jc)=0 (5.21) is, under certain assumptions, conditionally oscillatory (here hog0t := 1). This would generalize the result of [58] concerning generalized Euler-type equation (1.19) and give us the possibility to generalize the oscillation and nonoscillation criteria of the paper [66] to the case when we study perturbations of equation (5.21), similarly as in [20, 65], where perturbations of generalized Euler-type equation (1.19) are studied. Another task would be to find the asymptotics of nonoscillatory solutions of equation (5.21). Remark 5.3. Interesting problems arise from the approach of Fujimoto and Yamaoka ([68, 69]), who started to consider second-order half-linear equations with function p(t) instead of the number p in the operator <ř». A natural question from our perspective is to what extent the Riccati technique can be applied and if there could exist a variant of the modified Riccati transformation which would be of use. Remark 5.4. The qualitative theory of delayed and neutral half-linear equations is still a rather new research field where many branches deserve exploration. Compared to ordinary half-linear equations, one of the disadvantages is that, within the Riccati technique, only Riccati type inequalities can be applied. This fact leads to the bigger number of oscillation criteria, which are currently proved, whereas nonoscillation criteria are harder to achieve. -41- Conclusion_42 Remark 5.5. The applicability of the differential transform method for finding the numerical solutions of differential equations is limited by the proposed calculating algorithms in concrete programs like Matlab. To overcome the limits of extending the solution to longer intervals, a different approach to implementing the code could help. One of the possible directions to overcome the effect of division by a number close to zero might be the concept of automatic differentiation. 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Patikova, Asymptotic formulas for nonosdilatory solutions of conditionally oscillatory half-linear equations, Mathematica Slovaca, 60 (2010), pp. 223-236. 2. [110]: Z. Patikova, Nonos dilatory solutions of half-linear Euler-type equation with n terms, Mathematical Methods in the Applied Sciences, 43 (2019), pp. 7615-7622. 3. [65]: S. Fisnarova and Z. Patikova, Perturbed generalized half-linear Riemann-Weber equation - further oscillation results, Electronic Journal of Qualitative Theory of Differential Equations, (2017), pp. 1-12. 4. [66]: S. Fisnarova and Z. Patikova, Hille-Nehari type criteria and conditionally oscillatory half-linear differential equations, Electronic Journal of Qualitative Theory of Differential Equations, (2019), pp. 1-22. 5. [ 111 ]: Z. Patikova, Integral comparison criteria for half-linear differential equations seen as a perturbation, Mathematics, 9 (2021), pp. 1-10. 6. [67]: Z. Patikova and S. Fisnarova, Use of the modified Riccati technique for neutral half-linear differential equations Mathematics, 9 (2021), 1-13. 7. [114]: J. Rebenda and Z. Patikova, Differential transform algorithm for functional differential equations with time-dependent delays, Complexity, 2020 (2020), pp. 1-12. 8. [112]: Z. Patikova and J. Rebenda, Application of the differential transform to second-order half-linear Euler equation, submitted. Mathemalicci VERSITA Sl^^ DOI: 10.2478/sl2175-010-0008-8 Math. Slovaca 60 (2010), No. 2, 223-236 ASYMPTOTIC FORMULAS FOR NONOSCILLATORY SOLUTIONS OF CONDITIONALLY OSCILLATORY HALF-LINEAR EQUATIONS ZUZANA PATIKOVA ( Communicated by Michal Feckan ) ABSTRACT. We establish asymptotic formulas for nonoscillatory solutions of a special conditionally oscillatory half-linear second order differential equation, which is seen as a perturbation of a general nonoscillatory half-linear differential equation (r(t)$(x'))' + c(t)$(x) = 0, $0) = \x\P~1 sgnx, p > 1, where r, c are continuous functions and r(t) > 0. ©2010 Mathematical Institute Slovak Academy of Sciences 1. Introduction In this paper we investigate asymptotic properties of nonoscillatory solutions of a special conditionally oscillatory half-linear second order differential equation, which was constructed in [3] as a perturbation of a general half-linear differential equation (r(t)$(x'))' + c(t)$(x) = 0, = sgnx, p > 1, (1) where t £ [t0, oo), r, c are continuous functions and r(t) > 0. In the case p = 2, equation (1) reduces to the linear Sturm-Liouville differential equation (r(t)x'Y + c{t)x = 0 (2) 2000 Mathematics Subject Classification: Primary 34C10. Keywords: half-linear differential equation, conditionally oscillatory half-linear equation, modified Riccati equation. ZUZANA PÁTÍKOVÁ and it is well known that the linear oscillation theory of (2) can be naturally extended also to half-linear equation (1). In particular, (1) is called oscillatory if its every nontrivial solution has infinitely many zeros tending to infinity and nonoscillatory otherwise. In the whole paper we suppose that (1) is nonoscillatory. Let d{t) be a positive continuous function, we say that the equation (r(t)$(x')y + [c(t) + nd(t)]$(x) = 0 (3) is conditionally oscillatory if there exists a constant /io > 0 such that (3) is oscillatory for \i > /i0 and nonoscillatory for \i < /i0. Let h(t) be a positive solution of nonoscillatory equation (1) such that h'(t) 7^ 0 on some interval of the form [To, oo) and denote R(t) := r(t)h2(t)\h'(t)\p-2, G(t) := r(t)h(t)$(ti'(*)). (4) Under the asumptions di —— = oo, liminf \G(t)\ > 0, K[t) r->oo the authors of [3] constructed a conditionally oscillatory equation seen as a perturbation of (1) in the form (r(i)$(x'))' + c(t) + hP(t)R(t)(J R-^s) ds)2. $(x) = 0. (5) The critical oscillation constant of this equation is /io = where q is the conjugate number to p, i.e., ^ + ^ = 1. In [3] it is also shown that (5) has for this constant \x = /iq a solution with the asymptotic formula x(t) = h(t) í jR-1{s)ás\ (l + o(J^jR-1(s)ds^j as t —> oo. (6) The aim of this paper is to give more precise asymptotic formulas in terms of slowly and regularly varying functions in the case where the constant \x is less than or equal to 224 ASYMPTOTICS FOR CONDITIONALLY OSCILLATORY HALF-LINEAR EQUATIONS The "perturbation approach", when the studied equation is regarded as a perturbation of another half-linear equation, has been also used in [7]. Here, the asymptotics of nonoscillatory solutions of ($(x')y + ^(x) + c(tWx)=0, (7) where 7P = , was established under the assumption oo lim logt f c(s)sp~1 ds £ f-oo t—>oo J \ P Equation (7) has been seen as a perturbation of the half-linear Euler type equation ($(x'))' + %$(x) = 0 (8) with the critical constant jp. In this paper, we apply the perturbation principle combined with the so called Riccati technique to get our asymptotical results for (5) with /i < 2. Preliminaries As in the linear oscillation theory, the nonoscillation of equation (1) is equivalent to the solvability of a Riccati type equation (for details see [2]). In particular, if x is an eventually positive or negative solution of the nonoscillatory equation (1) on some interval of the form [To,oo), then w(t) = r(t)<& solves the Riccati type equation w' + c(t) + (p-l)r1~q(t)\w\q = 0. (9) Conversely, having a solution w(t) of (9) for t £ [T0,oo), the corresponding solution of (1) can be expressed as x(t) = Cexp jy r1_9(s)$_1(u;) ds where $_1 is the inverse function of $ and C a constant. Using the concept of perturbations it appears useful to deal with the so called modified (or generalized) Riccati equation. Let h be a positive solution of (1) 225 ZUZANA PATIKOVA and Wh{t) = r(t)$ (jt^) be the corresponding solution of the Riccati equation (9). Let us consider another nonoscillatory equation (r(t)$(x'))' + C(t)^(x) = 0 (10) and let w(t) be a solution of the Riccati equation associated with (10). Then v(t) = (w(t) — Wh{t))hp{t) solves the modified Riccati equation v' + (C(t) - c(t))hp +pr1-qhpP($-1(wh)Jw) =0, (11) where \u\p \v\q P(u, v) :=--uv H--> 0, P Q with the equality P(u,v) = 0 if and only if v = (0, oo) be such that J'(t) > 0 for t > T0, lim J(t) = oo and let g(t),e(t) be some measurable functions satisfying lim g(t) = g G (0, oo) and lim s(t) = g G M. According to the terminology of the above mentioned papers a positive measurable function f(t), such that / o J-1 is defined for all large t, and which can be expressed in the form /(*) = 9(t) exp || ds J , t > To for some T0 > t0, is called generalized regularly varying function of index g with respect to J (the notation / G RVj(g) is then used). If g(t) = g (a constant), f(t) is called normalized regularly varying function. For g = 0 the terminology (normalized) slowly varying function is used. 226 ASYMPTOTICS FOR CONDITIONALLY OSCILLATORY HALF-LINEAR EQUATIONS Let us only remark that in [6] this terminology is introduced for J(t) denned on the interval [0, oo) but since the point is in describing the asymptotic behavior, the interval of existence of the function f(t) is sufficient even for J(t). 3. Main results In this section we establish asymptotic formulas for nonoscillatory solutions of conditionally oscillatory equation (5) in the cases \i < ^ and \i = respectively. Theorem 1. Suppose that (1) is nonoscillatory and posseses a positive solution h(t) such that h'(t) ^ 0 for large t and let oo Jwr°°- (14) and liminf \G(t)\ > 0. (15) If \i < j^, then the conditionally oscillatory equation (5) has a pair of solutions given by the asymptotic formula (g-l)Ai Xi = h{t) I JR-l(s)ds\ Li(t), where Xi are zeros of the quadratic equation |a2-a + /i = 0 (16) and Li(t) are generalized normalized slowly varying functions of the form Li(t) = exp < J--ds > and Si(t) —>■ 0 for t —>■ oo. I fl(S)Jfl-l(T)dT J Proof. We are looking for solutions of the modified Riccati equation associated with (5), which reads as +--+ (p- iy-Ht)h-Ht)\G(t)\qF = o, (17) R(t)(fR-i(s)ds)i V U/ where G is denned in (4) and F in (13). 227 ZUZANA PÁTÍKOVÁ Assumptions (14) and (15) imply the convergence of the integral J r1-q{t)h-q{t)\G{t)\'iF i^-A dt, from which follows (see [3]) that v(t) -> 0 and —> 0 for t —> oo. Let Cq[T, oo) be the set of all continuous functions on the interval [T, oo) (concrete T will be specified later) which converge to zero for t —> oo and let us consider a set of functions V = {u e C0[T, oo) : \u(t)\ T}, where e > 0 is so small that 1 (g+l)e I. Therefore Flu)----u <(q- l)Mq\u\ (21) Similarly, F'{u) = F"(0)u+ *""({')..2 , ^-1)^-2)^ , , 2 -u = q(q—l)u+ ii+c,r3sgn(i+c/: u , where (' is between 0 and u. Again, considering \('\ < ^ we have \F'(u) - q(q - l)u\ < 3(q - l)Mq\u\2. (22) 229 ZUZANA PATIKOVA Now, let us denote J(t) := J R 1(s)ds, then the estimate of the function Ei{z,t) for z G V, reads as \Ei(z,t)\ \qz(t) + lj2{ť)vf{z, t) + {p-l)J2{t)G2{t)F 'vj(z,ty . G(t) . <\íi-\%-\iqz(ť) + l{\ + z{t))2\ + (p-l)J2(t)G2(t) F H(z,t)\ _ q{q - 1) /Vj(z,t) G(t) J 2 ^ G(t) . S2lZ[t)l + \G(t)\\J(t)\ S2lZ[t)l + \J(t)\ where (21) was used for w = ff and K := sup iGL is a finite constant for T t>T 1 1 n sufficiently large because of (15). According to (14) there exists 7\ such that the last term in the previous inequality is less than e2 and therefore \E%{z,t)\ Ti. Furthermore, for z\, z2 G V we have \Ei(zut) - Ei(z2,t)\ -\lq{z1-z2) + (p-l)J2{t)G2{t) F 'vi(zi,ty F 'vi(z2,ty G{t) j ~ v G(t) which, by the mean value theorem with a suitable z(t) G V, becomes -\iq(zi - z2) + qj(t)vi(z,t)(zi - z2) + (p- l)J(t)G(t)F' (^4^) (z, - z2) - qJ(t)vi(z, t){Zl - z2) G{t) < -Xiq + qJ(t)vi(z,t) + (p-l)J(t)G(t) (F'(^^-q{q-l) , i = 1, 2. [J R(s) J i2-i(r) dr J Then equation (20) is equivalent to (n(t)z(t))'+ n(t)--i-Ei(z,t) = o. (24) For i = 1, we have a function ri(t) = exp ■1 + Xiq R(t) fR-^s) ds ds > = exp (25) -y/l-2qn ds R(s) JR-Ht) dr R(s) J JR"1(r)dr and it is easy to see that r\{t) —> 0 for t —> oo. Finally, let us define an integral operator F\ on the set of functions V by {FlZ)(t) n(t) rns) We observe that ri(s i R(s)fR-1(r)dr ■Ei(z, s) ds. ds Jt R(s)fR-i(r)dT Taking T = max{Ti, T2}, by (23) and (18) we have y/l-2qn' \(Fiz)(t)\ < nis) n(t) \Ei(z, s) \ ds < { R(s)jR-1(r)dr 1 /q + l)e2 = exp < [-j——^-ds R(s)jR-1(r)dT J [/ R(s) J R-1 (t) dr and we define an integral operator F2 by t (F2Z)(t) = --±r í--£2(^,s)ds. T2[t)J R(s)jR-1(r)dr Since t r2(s) ds r2(t) - c R(s)JR-l(r)dr ^ 2qfi where c is a positive suitable constant, the inequality t 1 f r2(s) ds < r^t)J R(s)fR-i(T)dT V1 " ^ holds for t sufficiently large, as r2(t) —> oo for t —> oo. Taking T = max{Ti, T2}, the estimates for the operator F2 are the same as in the previous case and we can find a fixed point z2(t) satisfying F2z2 = z2. Thus z2{t) solves (25) and v2(t) = Xt2+Z2^ solves the modified Riccati equation (17). / R-1 ds 232 ASYMPTOTICS FOR CONDITIONALLY OSCILLATORY HALF-LINEAR EQUATIONS Expressing the solutions of the standard Riccati equation for (5) corresponding to the solutions Vi(zi, t) of the modified Riccati equation, we have Wi(t) = h-p(t)Vi(Zi,t) + wh(t) = wh(t) (l + mii i K + Zj(t) \ I Xj + Zj(t) wh{t) 1 H---- = wh(t) 1 H---- hP(t)wh(t) JR-tdsJ \ G(t)JR-1(s)ds, Since solutions of (5) are given by the formula x(t) = exp |J r1_g(s)$_1 (w) ds we need to express (\ q-i x { Xj + Zjjt) G(t)fR-i(s) ds, h'{t) (l + (q_ ^ K + Zj(t) +o( K+ Zi(t) k^ x G(t) J R-l(s)ds \G(t)jR-l(s)ds, h'{t) (q-l)Xi , (q-l)zi(t) ( Xi + Zi(t) H--t--1--1--h o k^ R(t) j R~1(s)ds R(t) j R~1(s)ds \R(t) J R'^s) ds, Because t R(t) J R-^s) ds o \i + Zj(t) M(t) J R~l(s) ds/ R(t) J R~l(s) ds = o(Xi + Zj(t)) ~ t R(t) J R-1^) ds holds for large t, the pair of solutions of (5) for i = 1, 2 is in the form Xi(t) = exp<^ \ogh(t) + log I R~1(s)ds t f (q-l)zi(s) + o(Xi + Zi(s)) ' R(s)}R-1(r)dT 233 ZUZANA PÁTÍKOVÁ As Zi £ V and hence Zi(t) —> 0 for t —> oo, the statement of the theorem holds for Si(t) = (q- l)Zi(t) + o(Xi + Zi(t)). □ Now let us present the asymptotic formula in case \i = which gives an improved version of (6). Theorem 2. Let the assumptions of the previous theorem be satisfied and let \i = Then equation (5) has a solution of the form where L(t) is a generalized normalized slowly varying function of the form L(t) = exp < J--ds > and e(t) —> 0 for t —> oo. I fl(S)Jfl-l(T)dT J Proof. For \x = ^ the quadratic equation (16) has a double root A = K We assume the solution of modified Riccati equation to be in the form (for z £ V) v(z,i) t J R-^s) ds which gives, after substituting into the modified Riccati equation (17) for \i = At) + R(t) J R-1^) ds + (p- iy-q(t)h-q(t)\G(t)\qF ^ t V{Z^ ÍR-Us)ds = 0. G(t) J Let us denote E(z,t) = -z(t)-j- + (p-l) (jR-l(s)ds\ G2(t)F^ G(t) . and let us consider an integral operator F3 00 (F3z)(t)= f-s-i-E(z,s)ds { R(s)jR-^r)dT on a set of continuous functions V = {u e C0[T, 00) : \u(t)\ T}, 234 ASYMPTOTICS FOR CONDITIONALLY OSCILLATORY HALF-LINEAR EQUATIONS where T and e are to be established similarly as in the proof of the previous theorem. Then the solution of modifed Riccati equation and also the solution of the studied equation can be found in almost the same manner as for the previous statement. □ p- Remark 1. If r(t) = l,c(t) = 7Pt p and h(t) = t p then the conditionally oscillatory equation (5) with perturbation of the Euler equa- tion (8), becomes the Euler-Weber (or alternatively Riemann-Weber) half-linear differential equation 1p Pp tP ' tP log2 t_ with the so-called critical coefficient \xp = \ \ ~p~) • ^he asymptotic formula (26) then reduces to the formula given in [7, Theorem 2]. Remark 2. For the Euler-Weber half linear equation also the asymptotic formula for its second linearly independent solution is known (see [8]). An open question remains whether the second linearly independent solution of (5) with \i = ^ could be found in a similar form = 0 p-i fMx'))' + x2(t) = h(t) / R-1(s)ds log / R-1(s)ds L2(t) where Mt)=exp /----^ds [J R(s)JR-^r)drlog ^JR-^r)drj and S2(t) —> 0 for t —> oo. REFERENCES [1] DOŠLY, O.—PATÍKOVA, Z.: Hille-Wintner type comparison criteria for half-linear second order differential equations, Arch. Math. (Brno) 42 (2006), 185—194. [2] DOŠLÝ, O.—ŘEHÁK, P.: Half-Linear Differential Equations. North-Holland Math. Stud. 202, Elsevier, Amsterdam, 2005. [3] DOŠLY, O.—UNAL, M.: Conditionally oscillatory half-linear differential equations, Acta Math. Hungar. (To appear). [4] HOWARD, H. C.—MARIC, V.: Regularity and nonosdilation of solutions of second order linear differential equations, Bull. Cl. Sci. Math. Nat. Sci. Math. 20 (1990), 85—98. 235 ZUZANA PÁTÍKOVÁ [5] JAROS, J.—KUSANO, T.—TANIGAWA, T.: Nonoscillation theory for second order half-linear differential equations in the framework of regular variation, Results Math. 43 (2003), 129-149. [6] JAROS, J.—KUSANO, T.—TANIGAWA, T.: Nonoscillatory half-linear differential equations and generalized Karamata functions, Nonlinear Anal. 64 (2006), 762—787. [7] PATIKOVA, Z.: Asymptotic formulas for non-oscillatory solutions of perturbed Euler equation, Nonlinear Anal. 69 (2008), 3281-3290. [8] PATIKOVA, Z.: Asymptotic formulas for solutions of half-linear Euler-Weber equation, Electron. J. Qual. Theory Differ. Equ. 15 (2008), 1-11 (Proc. 8'th Coll. Qualitative Theory of Differential Equations). Received 14. 5. 2008 Department of Mathematics Accepted 3. 8. 2008 Tomas Bata University in Zlín Nad Stráněmi 4511 CZ-760 05 Zlín CZECH REPUBLIC E-mail: patikova@fai.utb.cz 236 Received: 8 February 2019 Revised: 26 June 2019 Accepted: 2 September 2019 Check for updates DOI: 10.1002/mma.5930 SPECIAL ISSUE PAPER WILEY Nonoscillatory solutions of half-linear Euler-type equation with n terms Zuzana Patikova Department of Mathematics, Faculty of Applied Informatics, Tomas Bata University in Zlín, Nad Stráněmi 4511, 760 05 Zlín, Czech Republic Correspondence Zuzana Pátíková, Department of Mathematics, Faculty of Applied Informatics, Tomas Bata University in Zlín, Zlín, Nad Stráněmi 4511, 760 05 Zlín, Czech Republic. Email: patikova@utb.cz Communicated by: Z. Dia We consider the half-linear Euler-type equation with n terms tP jti íPLog/í IPLo&t |<ř(x) = 0, <ř(x) = |x|p Jsgnx in the subcritical case when 0l. The solutions of this nonoscillatory equation cannot be found in an explicit form and can be studied only asymptotically. In this paper, with the use of the perturbation principle, modified Riccati technique, and the fixed point theorem, we establish an asymptotic formula for one of its solutions. KEYWORDS asymptotic formulas, Euler equation, half-linear differential equation, nonoscillatory solution, perturbation MSC CLASSIFICATION 34C10 1 I INTRODUCTION The aim of the paper is to find an asymptotic formula for a nonoscillatory solution of half-linear Euler-type differential equation I <ř(x) = 0 P ' j=itPLog]t tPLog2nt/ for 0l, n e N, yp and fcp are the constants fp-iV i(p-i p-1 and Log/ are products of iterated logarithmic functions: j L°g/f: = niogfcf, iogit: = iogf> iogkt: = iogfc-i(iogo> k ^ 2. *:=i The studied Equation (1) is a special case of a general half-linear second-order differential equation L[x]: = (r(f)$(x'))' + c(f)$(x) = 0, (1) (2) Math Meth Appl Sei. 2020;43:7615-7622. wileyonlinelibrary.com/journal/mma © 2019 John Wiley & Sons, Ltd. I 7615 7616 Wiley PÁTÍKOVÁ where x=x(f), functions r(f),c(f) are continuous, and rif) is positive on the interval of consideration. The solution space of (2) is linear (but not additive) and contains either oscillatory or nonoscillatory solutions. Within the studies of Equation (2) in some neighborhood of infinity, ie, fe[T,oo) for some T, its solution can be classified as oscillatory, if it has got infinitely many zeros tending to infinity, and nonoscillatory otherwise. Oscillatory and nonoscillatory solutions cannot coexist; hence, half-linear equations are said to be oscillatory or nonoscillatory according to behavior of their all solutions (for more information, see the basic literature1 summing up the results for half-linear equations up to the year 2005). Describing asymptotic properties of solutions is one of the main tasks of qualitative theory of differential equations. Asymptotic behavior of solutions is often being classified with the use of the theory of regularly varying functions in the sense of Karamata. Let us recall the following notation (for more information, see, for example, the monograph2). A measurable function /:[a,oo)->(0,oo) is called regularly varying (at infinity) of index -9 (and we write fE.RV(d)) if If £=0, / is called slowly varying. The representation theorem (see, for example, Řehák3) says that /E.RV(&) if and only if it can be expressed in the form where t>a for some a>0, cp and ip are measurable functions such that limí^0O oo, where C < —^y. However, the results are inapplicable to our Equation (1), because the constant C in this case does not satisfy strict inequality, but equality. The reason is that Equation (1) lies on the threshold between oscillation and nonoscillation. For such equations, it was shown, for example, in Patikova7 that it can be useful to regard the studied equation as a perturbation of a nonoscillatory equation with less terms. Equation (1) can be seen as a perturbation and in some sense also a generalization of the half-linear Euler equation The critical coefficient y=yp is its oscillation constant: Replaced by a greater constant, the Euler equation becomes oscillatory, and for yp and smaller constants, it is nonoscillatory. In view of this property, one can say that the Euler equation is conditionally oscillatory. Equation (3) in the case y=yp has a pair of linearly independent nonoscillatory solutions (see, for example, Došlý and Řehák1, chapter 1.4.2) A for every A > 0. i coo Qc = f-1\t c(s)ds-C ((h(t)). (7) Then v(ť)=hp(ť)w(ť)— G(í) is a solution of the so-called modified Riccati equation (see, for example, Došlý and Fišnarová13) where v'(í) + h(t)L[h](t) + (p - iy-«(t)h-q(t)H(y, G) = 0, (8) WlLEY- PATIKOVA H(v, G) = \v + G\q - q$_1(G)v - \G\q. According to Došlý,9 Equation (8) is in some sense close to the so-called approximate Riccati equation u (t) + h(t)L[h](t) 2R(t) ■u2(t) = 0, (9) where the nonlinearity is only quadratic. The estimates of how close to each other are the solutions of (8) and (9) can be done with the use of the calculations from the proof of theorem 2 in Došlý and Ůnal14 (see also Došlý9, lemma 1), which we formulate here in the case of our interest for r(t)=l, h(t) = t ? , G=2fxp=Fp, R = I ^- J t. Lemma 2.1. If z£(0,l) and then for W/2[ip\3, (1-s)9"3 for q < 3, (P-I) ■H(v, rp)- (10) 3 I MAIN RESULT The main result of our paper providing an asymptotic formula of a solution of (1) reads as follows. Theorem 3.1. Equation (1) with [iE.(0,[ip) has a solution of the form 1 2A x(t) = t p Log^ilog^l + o(l)) as i^oo, (11) , i i /-jr where / = - + -,/1 — ■ 2 2Y np Proof We consider Equation (1) to be a perturbation of the Euler equation (3), which is the reason for choosing its solution as the function h in the modified Riccati equation (8). The modified Riccati equation (8) for (1) with h(t) =tp, G(t) = tp = 2^p reads as n - 1 ix "'(0+ E -rzz. ■h(v(í), rp) = o, jti fLogft tLog^t t and the approximate Riccati equation (9) is the in the form U(t) + "Z-^V + —^T" + — U2(t) = 0. (12) (13) (14) One can see that, with the use of the substitution u = 4(j.pt—, this is the classical Riccati equation of the linear second-order equation (4fipty'(t))' + ( + b(0 = 0, \j=1thogp tLogzntJ PATIKOVA ur„ 7619 Wiley- a which can be rewritten, using the notation r = —, as , , A"1 1 r \ (fy (0) + Z-t +-5- mO = o. (is) The transformation of variable ^df = ds, t = e?, s = logf leads to the equation First, we use the change of variable and substitution s = e"1, .y(s) = v^Zi(ui)> and if n>2, we continue in the same manner with ut = e"i+1, Zi(ut) = \fUiZi+i{ui+i) for i=2,...,n—1 to obtain the equation n x Z„_l("n-l) + J^—Zn-l("n-l) = 0. This equation has a couple of solutions zn-i(un-i) = uti-v where Alj2 are two (real) zeros of the quadratic equation A2 -A + - = A2 -i + JL = 0- (16) 4 4,up Backward transformations result into the couple of solutions of (15) y1>2(t) = v'Lo^(log„f)Al'2 and solutions of (14) are u -4u/ - Mi ^L°g"-lř l 2l-2l0g"ť (17) í=i Log;í ť ' Log„í Observe that for (iE(0,(ip), the zeros Alj2 of (16) are in the interval (0,1). The bigger one X\ = i + ^/l ^ lies in the interval (i l).Now, let us introduce the function 9(0 = P^d* (18) and the set of functions V = {v e C[T*, oo), |v(í)-ui(í)l < Kcp(t)}, where T and Kwill be specified later. Since oo and u^t) is positive for large t, there exists T0 such that u^t)—Kcp(f)>0 for í>T0. x To find a solution of the modified Riccati equation (13), we construct the integral operator '<•>"> = Í" gs^+sfe) *+■»* o» and show that it has a fixed point on the set V. t veV and t>T0, we have 00, we rcop — 1 First, we show that the integral J —-—H(v, rp)dt converges. Be 2\ such that the estimate (10) holds. For —Wl LEY- observe (using Lemma 2.1 and suppressing arguments) that for t > max{T0, Ti} PATIKOVA p-1 poop — 1 fO J ^—h(v, rp)dí < f < Lb(e) f°°—dí + f di ~~ f J 4[xpt ■h(v, rp)- < Lb(e)J typt -dt < Íoo 1 0 -v2dt typt -ydf < oo. Next, we show that for suitably chosen constants K and T, the operator 5~ maps V into itself. Using (19) together with (14), Lemma 2.1, and estimates of v, we have for t large enough l-oon — 1 1 i-ooli? |5-(v)(0-ui(0l = J, ^—H(v, rp)ds- — J, -^ < 1-ooD — 1 1 poo V2 jtP—H(v,rp)ds--Jtjás typ s foov3 , 1 fo )i , r°°u?, foo3iOi2ffl + 3^2Uioo. Finally, we show that 1 roo2UiK(d , „ -Jt-^dsT2- Now, inequality (21) implies JT^ds < 2|Up21J7(-2u1(s)u;(s))ds, PATIKOVA ur„ 7621 WlLEY- which is equivalent to T2-In total, we have |5-(v)(0-ui(0l < (Lbtf+Kh + oWMt). Let T3 be so large that the term o(l) is less than or equal to 1 for t>T3. Then for T* = max{T0, T1; T2, T3} and for K > ^(£) ^ 1 (such K exists since X\ e (-, 1)), we obtain 1 — Ai 2 \F(v)(t)-Ul(t)\^—^ = h~p(v + G), from which one can express a solution of (1) in the form x(t) = exp{j $_1(M;(s))ds}. In more detail, according to (12) and (17), Applying the inverse function -1, we have x'(t) p —11/ Logl ,t loglf „ „ NNX q y>-F 'l + f s"~1 +2^1—^- + 0(y(f)) x(f) p fV L°g„-if l°g„f and using the power^xpansion fprrnula, we arrive,at T,expansion formula, we arrive,at ■X =P^l + (q-l)t^+2tMq-l)pl + 0{cp{t)) :(0 P 'I Log„-i' l°gn* p-11 , lLog^t t 2Ailog„t , Q^(0 Finally, integrating gives (11). 4 I CONCLUDING REMARKS 1. Within the framework of regularly varying functions, we can say that the solution (11) is a regularly varying p — 1 3 function of index-(see the properties of indexes described, for example, in the appendix of Rehak ) and the function (l+o(l)) in its formula is a slowly varying function. 2. The so-called approximate Riccati equation (14) has two linearly independent solutions Ui,u2 described by (17). To get the solution of (13) and consequently of (1), we used ux—the one with the larger zero of (16). The natural question arises whether also u2 with the smaller zero of (16) could be used to find the second linearly independent solution of (1). This remains as an open problem. conflict of interest This work does not have any conflicts of interest. orcid Zuzana Pánková® https://orcid.org/0000-0003-1992-3895 references 1. Došlý O, Řehák P. Half-Linear Differential Equations North Holland Mathematics Studies, Vol. 202. Amsterdam: Elsevier; 2005. 2. Maric V. Regular Variation and Differential Equations, Lecture notes in Mathematics. Berlin, Heidelberg New York: Springer-Verlag,; 1926;2000. 3. Řehák P. Asymptotic formulae for solutions of half-linear differential equations. Appl Math Comput. 2017;292:165-177. https://doi.org/ 10.1016/j.amc.2016.07.020 4. Jaroš J, Kusano T, Tanigawa T. Nonoscillatory half-linear differential equations and generalized Karamata functions. Nonlinear Anal. 2006;64:762-787. https://doi.Org/10.1016/j.na.2005.05.045 5. Kusano T, Manojlovič JV. Precise asymptotic behavior of regularly varying solutions of second order half-linear differential equations. Electron J Qual Theo. 2016:62. https://doi.Org/10.14232/ejqtde.2016.l.62 6. Kusano T, Maric V, Tanigawa T. Asymptotics of some classes of nonoscillatory solutions of second order half-linear differential equations. Bull CI Sci Math Nat Sci Math 2003. 2003;28:61-74. https://doi.org/10.2298/BMAT0328061T 7. Pátíková Z. Asymptotic formulas for non-oscillatory solutions of perturbed Euler equation. Nonlinear Anal. 2008;69(10):3281-3290. https://doi.Org/10.1016/j.na.2007.09.017 8. Elbert Á, Schneider A. Perturbations of the half-linear Euler differential equation. Result Math. 2000;37:56-83. https://doi.org/10.1007/ BF03322512 9. Došlý O. Solutions of Riemann-Weber type half-linear differential equation. Arch Math. 2017;53(1):49-61. https://doi.org/10.5817/ AM2017-1-49 10. Pátíková Z. Some Remarks on Asymptotics of Nonoscillatory Solutions of Half-Linear Riemann-Weber Type Equations, to be published in AIP Conference Proceedings of ICNAAM; 2018. 11. Pátíková Z. Asymptotic formulas for solutions of half-linear Euler-Weber equation. Electron J Qual Theory Differ Equ Proc 8'th Coll Qualitative Theory ofDiffEqu. 2007;15:1-11. https://doi.Org/10.14232/ejqtde.2007.7.15 12. Hasil P. Oscillation and non-oscillation criterion for riemann-weber type half-linear differential equations. Electron J Qual Theo. 2016:59. https://doi.org/10.14232/ejqtde.2016.L59 13. Došlý O, Fišnarová S. Half-linear oscillation criteria: perturbation in term involving derivative. Nonlinear Anal. 2010;73(12):3756-3766. https://doi.Org/10.1016/j.na.2010.07.049 14. Došlý O, Ůnal M. Conditionally oscillatory half-linear differential equations. Acta Math Hungar. 2008;120:147-163. https://doi.org/ 10.1007/S10474-007-7120-4 How to cite this article: Patikova Z. Nonoscillatory solutions of half-linear Euler-type equation with n terms. Math Meth Appl Sci. 2020;43:7615-7622. https://doi.org/10.1002/mma.5930 Electronic Journal of Qualitative Theory of Differential Equations 2017, No. 69,1-12; https://doi.org/10.14232/ejqtde.20171.69 www.math.u-szeged.hu/ejqtde/ Perturbed generalized half-linear Riemann-Weber equation - further oscillation results Simona Fišnarová^1 and Zuzana Pátíková2 1Mendel University in Brno, Zemědělská 1, Brno, CZ-613 00, Czech Republic 2Tomas Bata University in Zlín, Nad Stráněmi 4511, Zlín, CZ-760 05, Czech Republic Received 14 July 2017, appeared 27 October 2017 Communicated by Mihály Pituk Abstract. We establish new oscillation and nonoscillation criteria for the perturbed generalized Riemann-Weber half-linear equation with critical coefficients Mx'))' + (7i + y -J±— + eft)\ d>(x) = o in terms of the expression —-- / ř(s)SP Log„slogn+1sds. The obtained criteria complement results of [O. Došlý, Electron. J. Qual. Theory Differ. Equ., Proc. 10'th Coll. Qualitative Theory ofDiff. Equ. 2016, No. 10, 1-14]. Keywords: half-linear differential equation, generalized Riemann-Weber equation, (non)oscillation criteria, perturbation principle. 2010 Mathematics Subject Classification: 34C10. 1 Introduction Consider the half-linear differential equation of the form L[x] := (r(ř)0(x'))/ + c(ř)0>(x) = 0, O(x) = \x\^1 sgnx, p > 1, (1.1) where r,c axe continuous functions, r(t) > 0 and t e [T,oo) for some T e R. The terminology half-linear comes from the fact that the solution space of (1.1) is homogenous, but generally not additive for p ^ 2. In the special case p = 2 this equation reduces to the linear Sturm-Liouville differential equation (r(t)x'Y+ c(t)x = 0. (1.2) In this paper we deal with oscillatory properties of equations of the form (1.1). It is well known that the classical linear Sturmian theory of (1.2) can be naturally extended also to (1.1), see [8]. 13 Corresponding author. Email: fisnarov@mendelu.cz 2 S. Fisnarovd and Z. Pdtikovd In particular, (1.1) is called oscillatory if all of its solutions are oscillatory, i.e., it has infinitely many zeros tending to infinity. In the opposite case all solutions of (1.1) are nonoscillatory, i.e., they are eventually positive or negative and (1.1) is said to be nonoscillatory. Let us emphasize that oscillatory and nonoscillatory solutions of (1.1) cannot coexist. If we suppose that (1.1) is nonoscillatory, one can study the influence of the perturbation c on the oscillatory behavior of the equation of the form (r(O(O(x')y + (c(t) + d(t))®(x) = 0. (1.3) The concrete (non) oscillation criteria measure the positiveness of the function c (generally of arbitrary sign). If c is "sufficiently positive" then the perturbed equation (1.3) becomes oscillatory, if c is negative or "not too much positive", then (1.3) remains nonoscillatory. This approach is sometimes referred to as the perturbation principle and leads, e.g., to the Hille-Nehari type (non)oscillation criteria for (1.3) which compare limits inferior and superior of certain integral expressions with concrete constants. These integral expressions are usually either of the form R-1(s)dsj^ c(s)hP(s)ds if J R-1(t)dt = oo (1.4) or /CO f't f'OO R-1(s)dsj c(s)ht>(s)ds if J R^tfdtKoo, (1.5) where h is a solution of (1.1) (or a function which is asymptotically close to a solution of (1.1)) and R = rh2\h'\P-2. Criteria of this type can be found in [1-3,5-7,9,10,13], see also the references therein. Note that the divergence or convergence of the integral J°° R_1(£) dt is closely connected with the so called principality of the solution h of (1.1), see [4,8] for details. Let us summarize the known results concerning the above mentioned criteria which apply to perturbations of the Euler and Rieman-Weber type equations. Denote An example of a nonoscillatory equation of the form (1.1) is the half-linear Euler type equation with the critical coefficient 7p (called also the oscillation constant) (cp(x'))' + ^cp(x)=0, (1.6) £-1 whose principal solution is h\(t) = t p1 arid the second one (linearly independent of hi) is asymptotically equivalent to hi(t) = t~ log? t, see [11]. Note that the criticality of yp in (1.6) means that if we replace 7p in (1.6) by another constant 7, then (1.6) is oscillatory for 7 > 7p and nonoscillatory for 7 < 7p. It was shown in [7] that the perturbed Euler type equation (O(x')y+(^ + c(0)o(x)=0 (1.7) is nonoscillatory if HmsupE(f) < jip, HminfE(f) > — 3}ip 1 t and oscillatory if liminf E(t) > jip, t^oa Perturbed generalized half-linear Riemann-Weber equation 3 where E(t) = logí Jř°° č(s)sP 1 ds. Došlý and Řezníčková [9] proved the same couple of non-oscillation and oscillation criteria with E(t) = fj č(s)sř'_1 log2 s ds. Compare both cases of E(t) with (1.4) and (1.5) taking h(t) = h(t) and h(t) = h2(t), respectively. Further natural step was to find similar statements also for perturbations of the Riemann-Weber (sometimes called Euler-Weber) half-linear equation with critical coefficients £-1 1 This equation has a pair of solutions asymptotically close to the functions h\(t) = t v log? t p-l 1 2 and h2(t) = t v log? ílogp(logí) and if we replace the constant Up in (1.8) by a different constant u, then (1.8) is oscillatory for u > Up and nonoscillatory for u < Up, see [12]. The (non)oscillation criteria for the perturbed equation + ( ^ + Jp^ + W ) = 0 (1-9) were formulated in terms of /CO c(s)sP-1logsds/ which complies with (1.4) taking h(t) = h\(t). The relevant nonoscillation criterion for (1.9) was proved in [2] and oscillatory criterion in [10]. The case which corresponds to (1.5) and to the second function h2 remained open. Recently, the criteria from [2,10] were generalized in [3] to perturbations of the following generalized Riemann-Weber half-linear equation with critical coefficients (*(*'))' + f S + t ) *M = °- (1-10) where n e N and log1í = logí/ logfcř = log^logí), k>2, Logyí = nj^=1logfcí. Elbert and Schneider in [12] derived the asymptotic formulas for the two linearly independent nonoscillatory solutions of (1.10). These solutions are asymptotically equivalent to the functions h(t) = Log př/ h2(t) = LogIřiog^+11. (l.ii) Došlý in [3] studied the equation LRW[x] : = (cD(x')y + ( ^ + £ + ~c{t) O(x) = 0 (1.12) and proved the following statement. 4 S. Fisnarovd and Z. Pdtikovd Theorem A. Suppose that the integral J°° c(f)fp 1 Logntdt is convergent, d) If limsup logn+11 / c(s)sp Lognsds -3up, then (1.12) is nonoscillatory. (ii) Suppose that there exists a constant j > ^^^f such that c(t)tv log3 t > jfor large t and liminflogn+1fy c(s)sP Lognsds > up. Then (1.12) is oscillatory. Observe that the integral expression from Theorem A relates to (1.4) with h(t) = h\(t) from (1.11). If n = 1, then (1.12) reduces to (1.9) and the criteria from Theorem A reduce to that obtained in [2,10]. The aim of this paper is to complement Theorem A (and also the corresponding results of [2,10] in case n = 1). We utilize the second function hi from (1.11) and find a related couple of criteria for equation (1.12) formulated in terms of the expression 1/^-1 2 —-- / c(s)SP Lognslogn+1sds lu6n+l1 J which corresponds to (1.5). 2 Auxiliary statements In this section we present the known statements which will be used in the proofs of our main results in the next section. Denote R(t) := r{t)h2{t)\h'{t)\v-2, G(t) := r{t)h{t)<5>{h'\t)) (2.1) and recall that q = -^-^ is the so called conjugate number of p. The following statement comes from [13]. Theorem B. Let h be a function such that h(t) > 0 and h'(t) ^ 0, both for large t. Suppose that the following conditions hold: /CO /"CO R_1(f)df -(-a-y/2a) (2.4) f->°o Jt Jt q for some a. > 0, then (1.1) is nonoscillatory. Perturbed generalized half-linear Riemann-Weber equation 5 The following theorem was proved in [6]. Theorem C. Let hbea positive continuously differentiable function satisfying the following conditions: h(t)L(h)(t) >0 for large t, / h(t)L(h)(t) dt = oo, (2.5) /CO R~1(*)d* = oo and lim G(*) = oo. (2.6) Then (1.1) is oscillatory. In the following lemma we summarize some technical facts which are either evident or were derived in [3]. Lemma 2.1. For n > 2 and large t we have Logn* > ■■■> Log^ = log* > • • • > logn* and 0og„ty i £-1 _ 1 Moreover, for h(t) = f~v~ Log,?* and the operator defined in (1.12) we /wroe and v ^-kog^fi+f: 1 M*)LRW [h](t) V Log^ * log3 * AS * —>■ 00. (2.7) 3 Main results Our main result concerning nonoscillation of (1.12) reads as follows. Theorem 3.1. If lim sup-—-—- / c(s)sP~1 Log ns\og2n+1s ds < 2}ip(-u + Via), t^oo logn+l t jt 1 ft _1 2 liminf---/ c(s)sp Log „s log„ ,! s ds > 2up(-a - V2a) t^oo logn+1*Jr n n+i p /or some a. > 0, */zen equation (1.12) is nonoscillatory. (3.1) (3.2) £-1 I 2 Proof. We prove the statement with the use of the function h(t) = t v Logn*log^+1 * in Theo- rem B. By a direct differentiation (and using Lemma 2.1) we have h'{t) p — 1 1 tIT \-i Log„* / 1 log* H----+ r?Lognp*lognp+1* + -* p Log? *-P P 1 2 tl 1 --1 1 + -,, Log„'Mog;+lt— ^ir* Logf (log|+1' U + E (^TijL^F + (P-l)Lcg„+l( L°gn* logn+l* 6 S. Fišnarová and Z. Pátíková Denote Tr p-i v p-i . Then = Tvr1+1v Logn"'tlogn;í Í 1 + e ■ 1 p-i + S(p-l)Logft (p-l)Logn+1^ By a direct differentiation (and using Lemma 2.1 again) we obtain 2 (*(&'))' = -7Pr2+? Logi""tio^;i t (i + e 1 p-1 + (p-l)Logf* (p-^Logn+i* +7pr2+?Log; 'tiog;+ííEL^(1 + I) i i 1=1 ^6l' + 27pí 2+"Logn^logn+íř 1 + E 1 + + 2 ^(p-l)Logfí (p-l)Logn+1^ \ p-i ftí (p-1) Logfí (p-l)LogM+1i p-2 + rp(p-i)ri+rLogn píiogn+1pí l + E + ,w (p-l)Logf* (p-^Logn+i* (p-l)i 1 1/1 1 + T-: h-: + log2í Log2ř Vlogř Log2ř7 LogMí^Logf 1 * 1 2 "+1 1 + --rE Logn+1f p Logft_ Observe that the expression in the square brackets can be rearranged as follows: 1 n -i Ett1i7+ E 2 "+1 1 + --rE ftí Logfí i<í . -7--1 i=i •-■"Si 1 + 7,ÉI^(i + E + i + E tí(p-l) Logfí (p-l)Logn+1í^ 1 2 + Logn+1í \ ftí (p-1) Logfí (p-l)Logn+1^ 1 ■ £ 1 2 1 l V ~ 1 i<£/<„ Lo§ Lo§ f V ~ 1 Log M+11 £i Log ft Next, denote p-2 «(*)== l + Er.-iM^, + ' ^(p-l)Log/* (p-l)Logn+1^ Using the power expansion s(s — 1) 9 s(s —l)(s —2) O . r. (1 + x)s = 1 + sx + -i——'-xl + —-^--x3 + o(xJ) as x ->■ 0, s e R, 2 6 we obtain , 2 Em_! | P~2 1 | 2 ^ I (P-2)(P~3) 1 1 j P - Lo^ LoW J 2(P " !)2 \k Log,-* 2 + 2(p-l)2 ^Log.-t Logn+1f (p - 2)(p - 3)(p - 4) / " _J_ 2 \ 3 3 6(p-l)3 ^Log.-t+Log^tJ +°llOS t)f as £ —>■ oo. Next observe that if at least one of the indices i, j, k is greater than one, then 1 oo. - = o (log t j as t ->■ LogffLogyfLog^ Hence we can write B(£) in the form B(f)=1+Ns^+^) (p-2)(p-3) /A 1 r 2 4 "+1 1 ' 2(P-1)2 LfeLog?* LogLogy* LogM+1tlti Log,-^ (p-2)(p-3)(p-4)^_ 3 + 6(p-l)3 log3* 1 8 j as * —>■ oo. From (3.3) and (3.4), we obtain A(t)-B(t) 2-p „ 1 2-p 1 n+1 1 + 7pp-l,JrO Log,tLog,t+ 7pp-l Log.. ,,t ^ (3.4) P " 1 i<£/<« Lo§ff Lo§/ P " 1 Lo§i=i Lo§ff p-2 n 1 fp-2\2 n 1 fp-2\2 1 ^^igl^+^i^ij glo^+^l^J Log^og/ 8 S. Fisnarovd and Z. Pdtikovd + 77^—?El—7-V V ~ 1 / Log M+1t ^ Log ft p - 1 log3 f p-2 1 /p-2\2 1 " 1 />-2 27P^—tt—-:+27P ^—r t——;L?r7+47 P-1L°S„+it ' "Vp-V LogM+1t,§ Logft ' VP-V Log2+1£ (p-2)(p-3) " 1 (p-2)2(p-3) 1 2(p-l)2 ^Log2f 'P 2(p-l)3 log3t (P~2)(p-3) r 1 (p-2)(p-3) 1 "+1 1 (P " !)2 i■ oo. Summarizing the above computations, we have WW - Log;-J(,og-f, (-7, - „ E -L^ - + .(log"' 0 as t —> oo. Consequently, for the operator Lrw defined in (1.12) we have hLRW[h] = h{$>{h'))' + hp(^ + t JL- + c(t)) Logntlog2n+1f f f ^_ 2TPP(p-2) 1 , -3,^ + tv-1 Log * log2 , t f ^ + r ^p - + c(t) (3.5) Log^log2^ / 27pKp-2) \ , 2 = nog3f V 3(p-l)2 +°(1)J+^ Logntlogn+1t as t —> oo. In order to check conditions (2.2), express R(t) and G(t) from (2.1): R(t) = h2(t)\h'(t)r2 p-2 f Lognnog2+1 f [ i + 7_TTT-_ + £Lognnog2+1£(l + o(l)) p y &" ^ £(p-l)Logft (p-l)Logn+1^ p_!\P-2 and G(t) = M*W(0) V Lognnog2+11 i + £ _m#w f + V ) bn &n+1 V f^(p-l)Log,i (p-l)LogM+1t p_!\P-i Lognnog2+1f(l + o(l)) Perturbed generalized half-linear Riemann-Weber equation 9 as t —> oo. Since f°° R 1(s) ds < oo, the first condition in (2.2) is satisfied. The second condition in (2.2) is also fulfilled, since 1 ds= f-^-V 2-^—(l + o(l)) (3.6) and hence R(s) \p-lj logn+1i r°° 1 v — 1 G(i) J ds = ^—LogMílogM+1í(l+o(l))->oo R(s) p as t —> oo. Finally, we show that conditions (2.3) and (2.4) hold. To this end, let e £ (0,1). Then iimLog„tlog;+lt oo. Conditions (3.1) and (3.2) together with (3.7) imply lim sup R 1(s)ds^j (^j h(s)LRW[h] (s) ds „ \p-2 lr1 1 I limsup---/ c(s)sP_1Lognslog2+1sds < -(-ot + Vla) and liminf R-1(s)ds^j ^j* h(s)LRVf[h](s) ds = (—?—) liminf-——- f c(s)sP-1Logns\og2n+1sds>-(-a-Vla). VP -1/ ^oo \ogn+1tJT bn bn+1 (*'))' + (^ + £ + d(t)) <*>(*) = 0, (3.8) \tP -=1 tP Log ■ t J which is in fact equation (1.12) shifted from n to n + 1. The reason why we formulate this criterion rather for (3.8) than for (1.12) is only technical. 10 S. Fisnarovd and Z. Pdtikovd d(t)tP\og3t>7>^J^^J (3.9) Theorem 3.2. Suppose that there exists a constant j such that .3,^ ^ 27pp(p-2) 3(p-l) for large t. If /CO dity-1 Log n+1t dt = oo, (3.10) then equation (3.8) is oscillatory. p-i i Proof. Take h(t) = t v Log^+1£. According to Lemma 2.1 (with n replaced by n + 1) h'{t) = ^r'L°gLt(1+°(1)) asf^oo. Hence, by (2.1) R(t) = ft2(t)l^(t)lP~2= f—V 2^Logn+1f(l + o(l)) asf^oo and consequently JR-1(s)ds=(^—-J logM+2t(l + o(l)) -> oo asf^oo. Further, G(t) = W(t))=(-—-)P LogM+1t(l + o(l)) ->■ oo asf^oo. P Rewriting (2.7) for the operator from (3.8) we have h(t)lRW[h](t) = flog t 27'P(2-P)+Wlog3t + O(l) 27Pp(2~p)+o(1) 3(P-1)2 £ log £ 3(p-l)2 as t —> oo. Because the integral J°° L°^n^ dt is convergent, condition (3.10) implies /co hlRW[h](t) dt = oo. Thanks to condition (3.9) we have also hI,Rw[h](t) > 0 for large t. This means that equation (3.8) is oscillatory by Theorem C. □ The following statement is the oscillatory criterion which complements Theorem 3.1. Theorem 3.3. Suppose that there exists a constant j such that tv w3 t (e(t)__to_^1 > 7 > 2^V-2) (3 n) for large t. If 1 I'1 W-1 L Lo§ «s lo§n+i s ds > UP' (3-12) c^ lu6n+l 1 J 1 then (1.12) is oscillatory. Perturbed generalized half-linear Riemann-Weber equation 11 Proof. Let us rewrite (1.12) into the form so (1.12) is seen as a perturbation of the generalized Riemann-Weber equation with the critical coefficients and with n + 1 elements in the sum. We apply Theorem 3.2 with the perturbation term d(t) = c(t) - tp^2 -. Then (3.9) is guaranteed by (3.11). With respect to (3.12) there exist e > 0 and f > T such that 1 ft liminf--- / c(s)sP Log ,,s log„ ,, s ds > up + e t^co logn+1 £ Jt + ^ and also / c(s)sP"1 Logn+1s logn+1 s ds > (pp + e) logn+11 for t > f. Let b > f. With the use of integration by parts and the above inequality, we have />~^1^^df-/rlLo^di At W-? ^Of"1 Logn+1* lo&,+l * df-^P [log„+2*]T lu6n+l1 1_ /% vp_iT , , f , ff JT£(S)SP lL°gn+lsl°gn+lS ds ^ +f: /;e(s)sPMlL°grf?+'sds d< - * ^ >t tLog„logn+1f ^ io^ /r C~(0fP" L0g tdt + K^ + If KoJ^i dt ~ V? ^n+21] \ > up + e + Ki + (^p + e) [logn+2 ~ - ^p [logn+2 t] bT = ^p + e + Kx + e logn+2 b - K2 ->■ oo as & —?> oo, where = ,r j^-.^ .log^.d, d[ ^= + T _ T Jt t Log „tlog2n+1t Hence condition (3.10) is satisfied and (1.12) is oscillatory according to Theorem 3.2. □ Remark 3.4. If a. = \ in Theorem 3.1, then 2up(—a + Via) = up, 2up(—u. — V2a) = —3up and the constants from (3.1) and (3.2) in Theorem 3.1 reduce to the constants in Theorem A, part (i). The generalization for a. ^ i is due to Theorem B. Note also that the constants in the nonoscillatory part of Theorem A could be generalized in the same way by utilizing [13, Theorem 3.2] in the proof of Theorem A. 12 S. Fišnarová and Z. Pátíková References [1] O. Došlý, A remark on the linearization technique in half-linear oscillation theory, Opus-cula Math. 26(2006), No. 2, 305-315. MR2272298 [2] O. Došlý, Perturbations of the half-linear Euler-Weber type differential equation, /. Math. Anal. Appl. 323(2006), No. 1, 426-440. MR2262216; https://doi.Org/10.1016/j.jmaa. 2005.10.051 [3] O. Došlý, Half-linear Euler differential equation and its perturbations, Electron. }. Qual. Theory Differ. Equ., Proc. 10'th Coll. Qualitative Theory of Dijf. Equ. 2016, No. 10, 1-14. 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Řehák, Half-linear differential equations, North-Holland Mathematics Studies, Vol. 202, Elsevier, 2005. MR2158903 [9] O. Došlý, J. Řezníčková, Oscillation and nonoscillation of perturbed half-linear Euler differential equations, Publ. Math. Debrecen 71(2007), No. 3-4, 479^88. MR2361727 [10] O. Došlý, M. Ůnal, Half-linear differential equations: linearization technique and its application, /. Math. Anal. Appl. 335(2007), No. 1, 450-460. MR2340334; https: //doi. org/10.1016/j.jmaa.2007.01.080 [11] á. Elbert, Asymptotic behaviour of autonomous half-linear differential systems on the plane, Studia Sci. Math. Hungar. 19(1984), No. 2-4, 447^64. MR874513 [12] á. Elbert, A. Schneider, Perturbations of the half-linear Euler differential equation, Results Math. 37(2000), No. 1-2, 56-83. MR1742294; https: //doi. org/10.1007/BF03322512 [13] S. Fišnarová, R Mařík, On constants in nonoscillation criteria for half-linear differential equations, Abstr. Appl. Anal. 2011, Art. ID 638271, 1-15. MR2846243; https: //doi. org/ 10.1155/2011/638271 Electronic Journal of Qualitative Theory of Differential Equations 2019, No. 71,1-22; https://doi.Org/10.14232/ejqtde.2019.l.71 www.math.u-szeged.hu/ejqtde/ Hille-Nehari type criteria and conditionally oscillatory half-linear differential equations Simona Fišnarová ^1 and Zuzana Pátíková2 1Mendel University in Brno, Zemědělská 1, Brno, CZ-613 00, Czech Republic 2Tbmas Bata University in Zlín, Nad Stráněmi 4511, Zlín, CZ-760 05, Czech Republic Received 18 February 2019, appeared 2 October 2019 Communicated by Josef Diblík Abstract. We study perturbations of the generalized conditionally oscillatory half-linear equation of the Riemann-Weber type. We formulate new oscillation and nonoscillation criteria for this equation and find a perturbation such that the perturbed Riemann-Weber type equation is conditionally oscillatory. Keywords: half-linear differential equation, generalized Riemann-Weber equation, non(oscillation) criteria, perturbation principle. 2010 Mathematics Subject Classification: 34C10. 1 Introduction In the paper we study oscillatory properties of the half-linear equation L[x] := (r(ř)0(x'))/ + c(ř)0>(x) = 0, O(x) = sgnx, p > 1, (1.1) where the coefficients r, c are continuous functions, r(t) > 0 on the interval under consideration, which is a neighbourhood of infinity. In the special case when p = 2 this equation becomes the linear Sturm-Liouville equation. If p ^ 2, equation (1.1) is called half-linear since it has one half of the properties that characterize linearity: the solution space is homogeneous, but is generally not additive. Despite the missing additivity, the classical linear Sturmian theory has been extended to half-linear equations. We refer to the book [8] for the overview of the methods and results concerning half-linear equations up to year 2005. Concerning the recent results on half-linear differential equtions, see, e.g., [14-20] and the references therein. Recall that equation (1.1) is said to be oscillatory if all its solutions are oscillatory, i.e., all the solutions have infinitely many zeros tending to infinity. In the opposite case equation (1.1) is said to be nonoscillatory. Note also that oscillatory and nonoscillatory solutions of (1.1) cannot coexist and this means that this equation is nonoscillatory if all solutions have constant sign eventually. Corresponding author. Email: fisnarov@mendelu.cz 2 S. Fisnarovd and Z. Pdtikovd Throughout this paper we suppose that equation (1.1) is nonoscillatory and we study the influence of perturbations of the coefficient c on the oscillatory behaviour of equation (1.1), i.e., we study equations of the form (r(f) Ao, we say that equation (1.3) is conditionally oscillatory with the oscillation constant Ao. Examples of conditionally oscillatory equations (written below with the oscillation constants) are the Euler type equation (O(x'))/+T|O(x)=0, 7p:=(Zj±y (1.4) and the perturbed Euler type equations, such as the Riemann-Weber type equation or equations with arbitrary number of perturbation terms of the form where n e N, log11 = log t, logfc t = logfc_1(log t), k > 2, Log = nJUi l°gfc ^- All these equations are nonoscillatory also in the critical case with the oscillation constants. The appropriate results concerning the Euler type equation and its perturbations in the coefficient including the asymptotic formulas for nonoscillatory solutions of these equations can be found in the paper of Elbert and Schneider [11]. Note that the result of Elbert and Schneider has been generalized to the case when also perturbations in the term with derivative are allowed and also to the case of equations with non-constant coefficients, see, e.g., [4,6,7,14] and the references therein. In this paper we study perturbations of general nonoscillatory equation (1.1). We suppose that h is a solution of (1.1) such that h(t) > 0 and h'(t) ^ 0, for t > to, where to is a real number from the interval of consideration of (1.1). Moreover, we suppose that /CO R-1(t)dt = oo and liminf \G(t)\ > 0, (1.7) where R(t) = r{t)h2{t)\h'{t)\P-2, G(t) = r(t)h(t)®(h'(t)). (1.8) Note that we follow the notation used in [10] and wherever we consider the integral J°°R_1(f)df, its lower limit is omitted, as it can be a constant greater or equal to to such that all relevant conditions hold. Hille-Nehari type criteria for half-linear equations 3 The motivation for our research comes from paper [10]. In that paper, under assumptions (1.7), it is shown that if d(t) = (hP(t)R(t)(f R_1(s) ds)2)-1 in (1.3), then (1.3) is conditionally oscillatory equation and the oscillation constant is ^, where q is a conjugate number to p, i.e., i + ^ = 1. It is also shown that the equation in the critical case with the oscillation constant ^ L[x]:=(r(0O(x'))' + c(f) +-;-? 2qhP(t)R(t)(f tR-1(s)ds)2 0 and x'(t) ^ 0, both for large t. Suppose that fCO x(t)L[x](t) dt is convergent and lim \G(t)\ [* J^- = oo, (2.2) t->°°' n Jt R(s) 4 S. Fisnarovd and Z. Pdtikovd where T e R is sufficiently large. If rt rCO J limsup / £_1(s) ds / x(s)L[x](s) ds < -{-a. + Via), (2.3) t^co Jt Jt q rt rco 1 liminf / R_1(s) ds / x(s)L[xl(s) ds > -(-a -V2a) (2.4) t^oo Jj Jt q for some a > 0, then (1.1) is nonoscillatory. Theorem B ([5, Theorem 1]). Let x be a continuously differentiable function satisfying conditions /CO x(t)L[x](t)dt < oo, (2.5) /■oo rff I _ , N = oo and limG(£)=oo. (2.6) r-t r-CO 1 liminf / R-1(s)ds / x(s)L[xl(s) ds > —, (2.7) t^oo Jt Jt 2q where T e R is sufficiently large, then (1.1) is oscillatory. Theorem C ([12, Theorem 3.1]). Let x be a function such that x(t) > 0 and x'(£) ^ 0, both for large t. Suppose that the following conditions hold: /co pCQ £_1(f)df -(-a -V2a) (2.10) /or some « > 0, T e R sufficiently large, then (1.1) is nonoscillatory. Theorem D ([5, Theorem 2]). Let x be a positive continuously differentiable function satisfying the following conditions: /CO x(t)L[x](t) dt = oo, (2.11) /co R-1(t)dt = oo and HmG(f) = oo. (2.12) Then (1.1) is oscillatory. In the next lemma we collect some technical facts which are frequently used in the proofs of our main results. Lemma 2.1. Suppose that conditions (1.7) hold, (i) Let j e Zbe arbitrary and k,l e Zbe such that k > 0, I > 0. Then log>'( f'R-1 (s)ds) , x lim —-^—'— = 0. 2.13 Gl(t)(ftR-1(s)ds)k Hille-Nehari type criteria for half-linear equations 5 (ii) The integrals G'(t)\ogi(ftR-Hs)ds) r V(/tR-1(s)ds) It G2{t)fR-1{s)ds Jt G{t)R{t){fR-1{s)dsy are convergent for arbitrary j e Z, T e R sufficiently large. (2.14) Proof, (i) Assumptions (1.7) imply that there exists a constant K such that G|t^ < K~ for sufficiently large t and , log>'( f'R-1 (s)ds) , x lim —-—-- = 0 2.15 t^oo (JtR-l(s)ds)fc which can be shown by L'Hospital's Rule as follows. If j < 0, then (2.15) is evident. If j > 0, then we apply L'Hospital's Rule j times to obtain logy*-1!.)*)*, _= 0. t^oo (JtR-l(s)ds)fc # t^°° (J'R-1(S)ds)k Therefore also (2.13) holds. (ii) The integrals are convergent by the comparison test for improper integrals. The first integral in (2.14) is convergent, because the integral G'(t) , 1 1 W d* = -r—- - lim G2(t) G(T) t^co G(t) is convergent and Hm V(/tR-1(s)ds) q t^00 JtR-1(s)ds Concerning the second integral in (2.14) we show that the integral V(/tR-1(s)ds) dt (2.16) It R(t)(fR-Hs)ds)2 is convergent. If j = 0, the convergence follows immediately from (1.7), since in this case 1 11 dt = —=--lim t R(t)(ftR-1(s)ds)2 JTR-1(s)ds ftR-1(s)ds' By induction, suppose that integral in (2.16) is convergent for a positive integer j and consider the case j + 1. Using integration by parts we obtain V+1(/tR-1(s)ds) = V+1(/TR-1(s)ds) _ ^ V+1(/tR-1(s)ds) t R(0(/tK-i(s)ds)2 JTR-1(s)ds JtR-1(s)ds + ; + 1 / it-——L Jt R(t)( tR-Hs)ds)2 This implies the convergence of the integral in (2.16) for any positive integer j. If j is negative, the convergence is evident. The convergence of the second integral in (2.14) follows then from the fact that G|t^ is bounded for large t. □ 6 S. Fisnarovd and Z. Pdtikovd In the last part of this section we evaluate xt[x] from (1.9) for some particular functions x. The first of the following statements comes from [10]. The identity (2.17) follows from [10, Theorem 3], where we use the fact that h'/h = G/R and fix the constant in the leading term. The convergence of the corresponding integral is shown in the proof of [10, Theorem 4]. It follows also from Lemma 2.1. Lemma 2.2. Let h be a positive solution of (1.1) such that h'(t) ^ 0 for large t and (1.7) holds. Set x(t) := h(t)(ftR-1(s)ds)'. Then {p-2){\-p)G'{t) 2p2G2(t) ftR-1(s)ds mm(t) = x:;/y-:^\{ a+°a)) as t —>■ oo and the integral +—2(i-p)(r2) 2d+0(D) 3V2G(t)R(t)(ftR-Hs)ds)2 x(t)L[x](t) dt (2.17) converges. In the proofs of the following two statements we use the notation q>(t) := J1 R-1(s)ds. (2.18) Lemma 2.3. Let h be a positive solution of (1.1) such that h'(t) ^ 0 for large t and (1.7) holds. Set x(t) := h(t) (/tR-1(s)ds)?logp (f R-1 (s) ds). Then x{t)L[x]{t) + 2qR(t)(ftR-Hs)ds)\oz(ftR-Hs)ds) (p-2)(l-p)G'(01og(/tR-1(s)ds) 2p2G2(t) f R-1(s)ds (l+o(l)) (2.19) + 2(l-p)(p-2)log(/tR-1(s)ds) + 3p2G(t)R(t)(ftR-Hs)ds)2 as t —> oo. Proof. We use notation (2.18). By a direct computation and using the fact that hG = h'R we obtain x'(t) = h'(t)cpr(t) log' cp(t) + h^R-'^cpp-'^logP cp(t) h'(t)q>v(t) log? cp(t) h'(t)cpr(t) log' f(t) ph'{t)R{t)(p{t) ph'{t)R{t)(p{t) log q>{t) 1 1 1+ + pG(t)(t) pG(t)q>(t)loS(p(ty Then r(t)(t))~^p~1(t) + (p-l)r(t)*(fc,(t))^(0(log9'(t))^^-2(tM,(t)- Consequently, where x(0(r(0O(x'(0))' = M0^-2(0B(0, B(t) := (r(t)(h'(t))) V(0 log ? (t)A(t) + ^r^M*))^*) log A(t) + P^!r(f)o(//(f))R-i(f)A(£) + (p - l)r(t)0(&'(t))p(t) logp(t)A'(t). P Next, for the derivative of A(t) we have G'i^cp^ + G^R-^t) A'(t) VG2(t)cp2(t) G'{t)(^(0))'log{h'{t)) log y(f) pR(t) p2G(t)R(t)cp(t) {p - l)(p - 2)r(t)0(ft'(t)) (p - l)rO(ft') (p - l)2r(t)0(ft'(t)) p*G(t)R(t)q>(t) pR(t) p2G(t)R(t)cp(t)\ogcp(t) (p - l)r(t)(h'(t))G'(t) log (p - l)r(t)(h'(t))G'(t) pG2(t) pG2(t) 8 S. Fisnarovd and Z. Pdtikovd Using the fact that G' = h(r<&(h'))' + h'r®(h') and hG = h'R, we simplify the previous formula as follows B(t) = (r(t) (®{h (t))) (ft'(f)))' (p - l)2r(f)(ft'(f)) log y(f) pG(t) p2G(t)R(t)cp(t) (p - l)(p - 2)r(t)*(fc'(0) (P " 1)M p2G(t)R(t)cp(t) p*G(t)R(t)q>(t) log ' To express AP~2(t) we use the power expansion (1 + x)s = £ ( S ) *''/ |x| < 1, s e R (2.21) with 1 1 + pG(t)p(f) pG(t)q>(t)loS(p(ty Note that the applicability of this power expansion is guaranteed by conditions (1.7). Hence Av-\t) = £ ( V 2 1 1 + pG{t)(ft'))' _ G' ti r®(h') _ ti G ~h~G~ti ~ W Hille-Nehari type criteria for half-linear equations 9 which follow from the definitions of R, G in (1.8), we get x(t)(r(tWx'(t)))'= HtW-2(t)B(t) = h{t){r{t)(<5>{h\t)))'cp{t)\ogcp{t) + + + + G(t)£^ log^ (pA^1 + ^-^-r^)R-x cpT log^ cpA^1 + 1V ~2r^^R-1 q'T log1-? cpA^1 + (p - l)^//')^ log^ loglog cp + (^(ft,)),logy + 2^(fe/)); + (p-lWfcQlogy pG pG pR (p - l)2rO(^)log

(h')G'log cp _ 2(p - l)r®(h')G' p2GRcplogcp pG2 pG2 M^OT)Viogy+(2'f'>(r('1>('''>>,'og'p + 2(2-'')(r('1>('''))' pG pG (p - l)2r<£>(h') log cp 2(p - 1)(2- p)rO(ft') 2(p - 1)(2 - p)rO(ft') p2GRcp p2GRcp p2GRcplogcp Next, since conditions (1.7) hold, we can use the power expansion (2.21) with 1 2 + pGcp pGcplogcp and we obtain AP-2 = i + l^l + 2(P~2) + (P-2)(P~3) + 2(p-2)(p-3) pG

2 log2

(h') (p-2)(l-p)(r®(h'))' + Gcp V p2 R- + 2P2 G 1 /2(p-l)(2-p)rO(^) 2(p-2)(l - p) (r®(h'))' G

(ft') | 2(p-2)(l-p)(r(ft'))' G(ft') + (p - 2)(1 - p)(p - 3) (r2 V V3 R 3P2 G Using the identities r(ft') _ h! (r®{h'))' _ G h1 R ~ h2' G ~h~G~¥' the above product Av~2B simplifies to | 1 /2(p-2)(l-p) G'\ | 1 /2(p-2)(l-p) G' G

(l+°(l)) + 3p2 G^2(l+°(l))- as £ —> oo. This means that xL[x] can be written in the form (2.22). □ 3 Oscillation and nonoscillation criteria for (1.10) The following theorem is an improved version of [10, Theorem 5]. In contrast to that result, we do not need the condition hmlog2 (f R-1(s)ds)R(t)G'(t) =0 considered in [10] and we have generalized the statement to oc ^ \. Theorem 3.1. Suppose that h is a positive solution of (1.1) such that h'(t) ^ Ofor large t, (1.7) holds and the integral f°°' g{t)hV{t) ftR~1(s)dsdt converges. If limsuplog ^ R-1{s)ds^j g{s)hP{s) J* R~V) drds < ^{-cc + \PIJk), (3.1) liminflog (J R-X{s)ds^ ^ g{s)h?{s) J* R~V) drds > J (-a - Via) (3.2) for some oc > 0, then (1.10) is nonoscillatory. 12 S. Fisnarovd and Z. Pdtikovd Proof. The idea of the proof is to apply Theorem A to equation (1.10), i.e., L := L. We take i x(t) = h(t) (J1 R_1(s) ds)v ■ By a direct differentiation and using the fact that h'R = hG, we get >(t) = h'(t)(ftR-Hs)dSy i + pG(t)JtR-^s)ds Now we express the functions R and G defined in (2.1) for this concrete x and use (1.8) and (1.7) to obtain R(t)=r(t)x2(t)\x>(t)r2 = r(t)h2(t)\h'(t)r2(ftR-Hs)ds) 1 + pG(t) JtR-1(s)ds p-2 (3.3) and = R(t)(ftR-1(s)ds)(l + o(l)) asf^oo G(t)=r(t)x(t)®(x'(t)) it)h(t)^(h'(t))(ftR-Hs)ds) 1 + pG(t) fR-H^ds = G{t)(f R-1{s)ds)(l + o{l)) asf^oo. It follows from (3.3) that p-i (3.4) J^R-1 (s)ds = (l + o(l)) log (Z^1) KeK, hence conditions (1.7) and (3.4) imply that (2.2) is fulfilled. Since x(t)L[x](t) = x(t)L[x](t) + g(t)\x(t)\P = X(t)U*](t)+g(tW(t) /V^ds, Lemma 2.2 and the condition for the convergence of f°°g(t)W(t) f R~1(s)dsdt guarantee that the integral f°° x(t)L[x] (t) dt is convergent and we have K_1(s) ds / x(s)L[x](s) ds -log ^Jtr-1{s)ds^j J™ (x(s)t[x](s)+g(s)W(s) J'r-1 {r)dr^j ds as t —> oo. Now we show that lim log ^ J1r'1 (s) ds^j xt [x] (s) ds = 0. By (2.17), it is sufficient to show that (3.5) and limlog / R-Hs)ds)f G(s)R(S)(/sR-i(r)dr)2 G'(s) ds = 0 G2(s) f R-^r) dr ds = 0. (3.6) (3.7) (3.8) Hille-Nehari type criteria for half-linear equations 13 Since linif^oo ff°° —-tf—ttt ds = 0, using L'Hospital's rule and (2.13) we have Jt GR(J R-1)2 o i limlogf / R_1(s)ds) / / w „,-—--ds t^°° 6Vi W /it G s )R s) TR-1 t dr 2 lim G(s)R(s)(fR-i(T)dT)2 .-2/ _ log-^/1 R-1 (s) ds) (f R-1 (s) ds)-^-1 (t) G(t) f R-1(s)ds hence (3.7) holds. To show (3.8), we use integration by parts ^ ^ ds =-7 ~--/ ——— ^——--— ds, G2(s)fR-i(r)dr G{t) f R-\t) dt Jt G(s)R(s)(f R-1(t) dr)2 which, together with (2.13) and (3.7), yields to (3.8). Hence (3.6) is proved. Consequently, by (3.5), we obtain J R^(s) ds J x(s)L[x](s) ds -log f^J1 R-1{s)ds^j J™ g{s)W{s) J" R^i^drds (3.9) as t —> oo. This means that conditions (2.3), (2.4) follow from (3.1), (3.2). All the assumptions of Theorem A are fulfilled, hence (1.10) is nonoscillatory. □ The next statement is an oscillatory counterpart of Theorem 3.1. Theorem 3.2. Suppose that h is a positive solution of (1.1) such that h'(t) ^ Ofor large t, (1.7) holds, the integral f°°g{t)W{t) f* R~l{s) dsdf converges and let there exist constants ji and 72 such that g(t)h?(t) ftR-1(s)ds >-71 -+-?2- (3.10) 7 " Gi(t)fR-Hs)ds \G(t)\R(t)(ftR-Hs)dsy for large t, where T^^f^sgnG', 72>2(^r2>sgnG. (3,1) liminflog ^ R-1{s)ds^j g{s)h?{s) J* K_1(t) drds > i- (3.12) then (1.10) is oscillatory. Proof. We apply Theorem B with L := L. Taking x(t) := h(t)(ftR^1(s)ds)p we obtain (3.3) and (3.4). Consequently, conditions (1.7) imply that both conditions in (2.6) are satisfied. Similarly to the proof of Theorem 3.1, we conclude that the second condition in (2.5) holds due to Lemma 2.2, since x(t)L[x](t) = x(t)L{x](t)+g(t)hP(t) f R-1(s)ds 14 S. Fisnarovd and Z. Pdtikovd and condition (2.7) follows from (3.9) and (3.12). Concerning the first condition in (2.5), we have from Lemma 2.2 that ww)= ^-m7p)g\t) (1+0(1)) 2p2G2(t)f tR-1(s)ds +—2(i-p)(r2) 2d+0(D) + g(t)fc"(t) /V^ds as £ —?> oo. Hence, the first condition in (2.5) is ensured by (3.10). Equation (1.10) is oscillatory by Theorem B. □ In the next theorem we handle equation (1.10) in the case, when the perturbation g(t) is of the form g{t) := -,----—,-, AeR. (3.13) kP(t)R(t)(f R-Hs)ds)2log2!/R-i(s)ds) In this special case equation (1.10) becomes conditionally oscillatory. Theorem 3.3. Suppose that h is a positive solution of (1.1) such that h'(t) ^ 0 for large t and (1.7) holds and consider the equation (r(f) ^ and there exists a constant j such that -5-7^-> 7^'^[ 7>— sgnG'(t) (3.15) R^log^fR-^^dr) - G2(t) p 6 w holds for large t, then (3.14) is oscillatory. Proof. If A ^ then the statement follows from Theorem 3.1 and Theorem 3.2. Indeed, if g(t) is given by (3.13), then /■oo ft fOO 1 / g{t)hP(t) R-\S)dsdt= -,- , ,-dt Jtskj KJJ Jt R(f)/tR-i(s)dslog2(/tR-1(s)ds) A , A hm log(/T R-1 (s) ds) log( ff R-1 (s) ds)' so the integral f°°g{t)W{t) ftR~1(s)dsdt is convergent. Consequently, concerning conditions (3.1), (3.2) and (3.12), we have lim log ^ J1 R-1 (s) ds^j J™ g{s)W(s) ^^ R-1 (t) dr^) ds = A. Hence, if —^ < A < then (3.14) is nonoscillatory by Theorem 3.1, where we take a. = ^ in (3.1) and (3.2). If A < — ^, the nonoscillation of (3.14) follows form the well-known Sturm comparison theorem. If A > ^, we use Theorem 3.2. It remains to show that condition (3.15) Hille-Nehari type criteria for half-linear equations 15 is sufficient for (3.10). Since A > ^, there exists e > 0 such that A = ^ + e and condition (3.10) with g(t) defined in (3.13) can be written as l 2q + £ > 7i\G[(t)\ | 72 R(t) log2!/ R-i(s) ds) G2(t) |G(t)|K(t) /' R-i(s) ds' where 71,72 satisfy (3.11). If we set 71 := ^, then (p-2)sgnG'(t) _ (p-l)(p-2) 7i > and condition (3.15) implies that 1 2p2 7i|G'(Q| 2qR(t)\og2(ftR-1(s)ds) ^ G2(f) Hence, it remains to show that 72 > sgnG' for large t. > i.e., log2(/tR-i(s)ds) \G(t)\fR-Hs)ds' 72log2(/tR-1(s)ds) e > \G(t)\JtR-Hs)ds for large t. This inequality is satisfied for any 72 6 K since the limit of the function on the right-hand side of this inequality is zero as t —> 00. Oscillation of (3.14) follows from Theorem 3.2. In the remaining part of the proof we deal with the critical case A = Tq- We use Theorem A withx(f) := fc(f)(/'r-^s) ds)hogp (ftR-1(s)ds) and L(t) := L(t) + 2qhP(t)R(t) (/' R-i (s) ds)2 log2 (/' R-i (s) ds) In this case we have from (2.20) (3.16) x'(t) = //,(0(/tR"1(s)ds)hogp(/tR-1(s)ds) 1 1 + + pG(0/tR-1(s)ds pG(0/tR-1(s)dslog(/tK-1(s)ds) hence, according to (2.1) (suppressing the arguments) we have R = rh2\h'r2(ftR-i)\ozUtR-1) 1 + + pGifR-1) PG(/fR-i)log(/fR-i) p-2 R(/tR-1)log(/tR-1) (l + o(l)) asf^oo and G = rh<&(h') (j1 R-1) log (J' R ft JD-V 1p-1 1 + + PG^R-1) peg4 R-^log^'R-1) rM>(//') (/' R"1) log(/f R-1) (1 + o(l)) as t -> 00. (3.17) 16 S. Fišnarová and Z. Pátíková These computations and conditions (1.7) imply that JtR-1{s)ds ~ log (^\og J1 R-1{s)ds^j as t -> oo (3.18) and thus condition (2.2) is satisfied. Next, since = q, from (3.16) and Lemma 2.3 we obtain x(t)L[x](t) = x(t)L[x](t) + 2í?R(ř)(/řR-1(s)ds)log(/řR-i(s)ds) G'(t)\oS(JtR-1(s)ds) np-2)(l-p) G2(t)jt(R-Hs)ds) 2p 2 + o(l) (3.19) + log(/iR-1(s)ds) '""''^^ r2(p-2)(l-p)+o(1) G(t)K(t) f/R-H^ds 3p2 as t —> oo. By Lemma 2.1 we have that f°° x(t)L[x](t) dt is convergent. Using L'Hospital's rule we obtain log(f (R-x(T)dr)) G(s)R(s)(fR-i(T)dT)2 log(/řR-1(s)ds)G-1(ř)K-1(ř)(/íK-1(s)ds)-2 lim log (log ( f'R-Hs)*)) r '^^f^Vs t^oo 6V V-/ G(s R(s )(fsR-i(T dr 2 = lim---f-^--—--—-——j---7--- (3 20) - log"2(log(/ř R-i (s) ds)) log"1 (Jř R-i(s) ds) (Jř R-i(s) ds)-^ (t) = Hm log2(log(/t R-X(s) ds)) log2(/t R-i(s) ds) = Q ř™ G(ř)/íR-1(s)ds Integration by parts gives G'(s) log(f R-1(t) dr) dg = log(/tR-1(s)ds) + r~ 1 - log(f R-i(T) dr) ^ G2(s)f (R-i(r)dr) G(ř) /řR_1(s) ds Jt G(s)R(s)(f R-i(r) dr)^ hence, by Lemma 2.1 and (3.20), we obtain lim log (log (/V(s)ds^ r^(s)log(rR-1(r)dr) 8V V-/ //•/* G2(s)f(R-1(r)dr) ^ ' Consequently, (3.20) and (3.21) imply rt roa / r-t \ i>oo lim / Ř_1(s) ds / x(s)L[x](s) ds = lim log log / R_1(s)ds) / x(s)L[x](s) ds = 0. t^oajT Jt t^oa \ J J Jt Hence, conditions (2.3) and (2.4) are satisfied with a = | and this means that (3.14) with A = is nonoscillatory by Theorem A. □ The next result is a nonoscillatory criterion for (1.10) based on Theorem C. Theorem 3.4. Let hbe a positive solution of (1.1) such that h'(t) ^ Ofor large t and (1.7) holds. If f'g(s)hP(s)(fsR-1(r)dr)\og1(fsR-1(r)dr)ds 1 ^ / x lim sup jTSy '—-—f-; 6 VJ-—-- < - (-a. + Via), 3.22 t^coť log(/íR-1(s)ds) q lim inf iTSi^ns^ÍR-Hr) dr) log2 (f R'1 (r) dr) ds > 1 ™~ log(/íR-1(s)ds) 0, T e R sufficiently large, then equation (1.10) is nonoscillatory. Hille-Nehari type criteria for half-linear equations 17 Proof. Take x(t) = h^if1 R_1(s) ds) v log' (/' K_1(s) ds) and L := L in Theorem C. Using (2.1) and (2.23) we express (suppressing the arguments) R = rh2\ti\P-2(ftR-^lo^if1R-1) [l+ i . + . t 2 . t . "T 2 = R(/tR-1)log2(/tR-1)(l+o(l)) asf^oo and 1P-1 1 2 pet/"!*-1) pG(/fR-i)log(/fR-i)_ = G(/' R-1) log2 (/' R-1) (1 + o(l)) as t -> oo. From these formulas we have that the integral J°° R_1(£) d£ is convergent since G = r&---(l + o(l))ds t R(s) Jt JsR-1(T)dTlog2(/sR-1(T)dT) = i-ctI i,^ (1 + o(1))<~- \ogfTR-i(t)dt Next, let us observe that \G(t)| f ^ ds = \G(t) | (f R-\s) ds) log (f R-\s) ds) (l + o(l)) -> oo as £ —> oo, hence conditions in (2.8) hold. Further we are interested in the expression I R^(s) ds J x(s)L[x](s) ds (x(s)L [x] (s) + g(s)hP (s) (f R-1 (t) dr) log2 (f R_1 (r) dr)) ds log(/tR-i(s)ds) as t —> oo. Since, by Lemma 2.1 and (2.22), the integral xL[x] ds is convergent, property (3.22) is sufficient for (2.9) and (3.23) is sufficient for (2.10). □ To formulate the oscillatory version of Theorem 3.3 we first prove the following oscillation criterion. Theorem 3.5. Suppose that there exist constants 71,72 satisfying (3.11) such that g(t)h?(t) [tR-1(s)ds >-7ll5(f)l-+-^- (3.24) 7 " G2(t)jtR-Hs)ds \G(t)\R(t)(jtR-Hs)ds)2 for large t. If /"«0*'(0(/'R-IWd«)log(/'K-'(.)d.)d. = », (3.25) then equation L[x] := (r(t)^(x'))' + (c(t) +--.- (3.26) V 2^(f)R(0(/tR-1(s)ds)2 1 +g(f)) 7i|G'(t)l + 72 G2(t)fK-i(s)ds \G(t)\R(t)(f R-i(s)ds); for large t. If liminf fTS(s)hns)(fR-HT)dT)\og2 (fR^^dr) ds > 1_ ™~ log(/tR-1(s)ds) V then equation (1.10) is oscillatory. Proof. Equation (1.10) can be rewritten in the form of (3.26) with 1 (3.28) g(t)=g(t) 2^P(t)R(t)(/t R-1(s)ds)2 log2 (/' R-1(s)ds)' on which we apply Theorem 3.5. Condition (3.24) follows from (3.27). Next we show that (3.25) holds. From (3.28) we have that there exists e > 0 and f > T such that ITg(s)hP(s)(fsR-Hr)dr)\oS2(rR-Hr)dr)ds ^ 1 f>f log(/tR-1(s)ds) 2q ' (3.29) Hille-Nehari type criteria for half-linear equations 19 Let b > T, then (suppressing some unnecessary arguments) 1 7: h {g(t) 2^(t)R(/tR-i)2log2(/tR- 2(?R(/fR-i)log(/fR-i) dt log(J R-1) With the use of integration by parts and the notation 2 - + e + Rx T 2 oo. □ 4 Remarks Remark 4.1. Let us consider the nonoscillatory Euler type equation with the oscillation con- £-1 stant (1.4). It is known that the function t v is a solution of this equation. To show how our results apply to perturbations of (1.4), consider the interval [1, oo) and take the solution h(t) £-2 v ' t v , p — 1 Then R(t) = t, G = ^-y- and ft R x(s) ds = log t, hence conditions (1.7) hold. Consequently, 1 _ Hp 2qhP(t)R(t) (ft R-1 (s) ds)2 tP log2 t 20 S. Fisnarovd and Z. Pdtikovd and this means that equation (1.9) becomes the Riemann-Weber type equation (1.5) and results of the previous section reduce to those for the perturbed Riemann-Weber type equation (*(^))' + + = o. (4.1) In particular, Theorem 3.1 with oc = | reduces to [1, Corollary 2], see also [2, Theorem 3.3] in the case n = 1. Theorem 3.2 is a generalized version of [9, Corollary 1] and also of [2, Theorem 3.3] with n = 1. Note that, since G' = 0, condition (3.10) simplifies to which is condition (3.15) from [2]. Concerning Theorem 3.3, observe that condition (3.15) is satisfied and equation (3.14) with A = ^ is equation (1.6) with n = 2. Hence, Theorem 3.3 generalizes results of [11] for n = 2. Finally, Theorem 3.4 applied to (4.1) is [13, Theorem 3.1] in the case n = 1. Remark 4.2. Based on the results of this paper and their comparison with those for the perturbed Euler type equation discussed in Remark 4.1, we suppose that we can study perturbations of equation (3.14) with A = ^ and find a perturbation such that the obtained perturbed equation is conditionally oscillatory. More generally, we conjecture that the equation with arbitrary number of iterated logarithmic terms (0(*'))' + I c{t) + Y-,---—- |cp(x)=0 (4.2) 1 /e02^(0R(0(/tK-1(s)ds)2Log2(/tR-i(s)ds) 1 is conditionally oscillatory (here Log0£ := 1). This would generalize the result of [11] concerning equation (1.6) and give us the possibility to generalize the oscillation and nonoscillation criteria of this paper to the case when we study perturbations of (4.2), similarly as in [2,13], where perturbations of (1.6) are studied. Remark 4.3. Let us comment the particular choice of the functions x in the proofs of our results. Consider the operators L and L defined in (1.9) and (1.10), respectively. If x is a solution of (1.9), then x(t)L[x](t) = g(t)xP(t) and hence, when applying one of the Theorems A, B, C to equation (1.10), the expression x(s)L[x](s) appearing in conditions (2.3), (2.4), (2.7), (2.9), (2.10) is replaced by g(s)xP(s). It has been shown in [3,10] that equation (1.9) has a pair of linearly independent solutions that are asymptotically close (as t —> oo) to the functions x1(t)=h(t) y R-i(S)dSy, x2(t) = h(t) ryntR-1(s)dsVlogp (jtR-1(s)ds We have taken x := X\ in the proofs of Theorem 3.1 and Theorem 3.2 and x := x2 in the proof of Theorem 3.4. Computations in proofs of these theorems together with Lemma 2.2 and Lemma 2.4 show that in both the cases x := X\ and x := x2, the expression xL[x] is small enough such that it does not have an influence on the limits superior and inferior in conditions Hille-Nehari type criteria for half-linear equations 21 (2.3), (2.4), (2.7), (2.9), (2.10), hence the expression g(s)x[(s) appears in (3.1), (3.2) and (3.12), and g(s)x2(s) appears in (3.22) and (3.23). In the proofs of Theorem 3.3 and Theorem 3.5 we have taken x := h{fl R-1) p log? 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Lett. 79(2018), 146-154. https://doi.Org/10.1016/j . ami.2017.12.012; MR3748624 mathematics ÍMDPIJ Article Integral Comparison Criteria for Half-Linear Differential Equations Seen as a Perturbation Zuzana Pátíková Department of Mathematics, Tomas Bata University in Zlín, Nad Stráněmi 4511, 760 05 Zlín, Czech Republic; patikova@utb.cz; Tel.: +420-57-603-5005 Abstract: In this paper, we present further developed results on Hille-Wintner-type integral comparison theorems for second-order half-linear differential equations. Compared equations are seen as perturbations of a given non-oscillatory equation, which allows studying the equations on the borderline of oscillation and non-oscillation. We bring a new comparison theorem and apply it to the so-called generalized Riemann-Weber equation (also referred to as a Euler-type equation). Keywords: half-linear differential equation; oscillation criteria; modified Riccati technique; Euler-type equation; second-order differential equation msc: 34C10 check for updates Citation: Patikova, Z. Integral Comparison Criteria for Half-Linear Differential Equations Seen as a Perturbation. Mathematics 2021, 9, 502. https://doi.org/10.3390/ math9050502 Academic Editor: Ravi P. Agarwal Received: 1 February 2021 Accepted: 24 February 2021 Published: 1 March 2021 Publisher's Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. [(g) 0 Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1. Introduction In this paper, we continue our research on Hille-Wintner-type comparison criteria for half-linear, second-order differential equations and provide an answer to one of the open problems stated in [1]. We study the equation of the form l[x] := (r(t)®(x'))'+ c(t)®(x) = 0, <$>(x) = \x\P~1 sgnx, p > 1, (1) where r, c are continuous functions and r(t) > 0. Equation (1) can be seen as a generalization of the second-order linear Sturm-Liouville linear equation, to which it reduces for p = 2, and it is well-known that many techniques for linear equations work effectively for half-linear equations too. Recall that one of the differences between half-linear and linear equations is well-visible in the notation—the attribute "half-linear" refers to the fact that the solution space of (1) has only one of the two linearity properties, where it is homogenous but not additive. On the other hand, classification of solutions and equations in terms of oscillation remains the same—a solution is called oscillatory if it has got infinitely many zeros tending to infinity, and non-oscillatory otherwise; and since oscillatory and non-oscillatory solutions cannot coexist, equations are classified as oscillatory or non-oscillatory according to their solutions. To refer to the most current results of the oscillation theory of (1), let us mention, for example, papers [2-6]. Because we are interested in the qualitative behavior of solutions of (1), we study it on a neighborhood of infinity, that is, on intervals of the form t > tg, where tg is a real constant. By saying that a condition holds for large t, we mean that there exists such an interval-neighborhood of infinity, where the condition holds. In our research, we focus on comparison theorems which compare two equations and their oscillatory properties. Let us consider another half-linear equation L[x] := (r(t)(<£>(x')Y + C(t)®(x) = 0. (2) Comparing the coefficient functions c(t) and C(t) (and even r(t) and its counterpart R(t) in general) pointwise, leads to the Sturm comparison theorems, whereas comparing Mathematics 2021, 9,502. https://doi.org/10.3390/rnath9050502 https: / /www.mdpi.com/journal/mathematics Mathematics 2021, 9,502 2 of 10 integrals with coefficient functions aims at Hille-Wintner-type criteria. In formulation of classical Hille-Wintner criteria for half-linear equations, one distinguishes two cases, depending on the behavior of the integral J°° r1-^ (t) dt. In case of its divergence and under the assumption that j C(t) dt < oo, the criterion says that if /CO rco c(s) ds < J C(s) ds for large t and (2) is non-oscillatory, then (1) is non-oscillatory too, see [7] or [8] Section 2.3.1. If the integral J°° r1_^(f) dt converges, denote p(t) := f(°° r1_^(s) ds and suppose that c(t) > 0, C(t) > 0 for large t. If /CO fCO c(s)pp(s)ds< J C(s)pp(s) ds < oo for large t, then non-oscillation of (2) implies non-oscillation of (1) (see [9] or [8] Section 2.3.1). Inspired by these results, in paper [1] we adopted the view of the perturbation principle, which allows to refine the results on the threshold between oscillation and non-oscillation, and proved the generalized version of the Hille-Wintner criterion in the following setting. Together with (1) and (2), let us have the equation of the same form L[x] := (r(t)(®(x')y + c(t)®(x) = 0, (3) which is supposed to be non-oscillatory, and let h be its positive principal solution. Equations (1) and (2) can be seen as perturbations of (3). The main result of [1] showed for the case J°° r1 ~1(t) dt = oo that under certain assumptions (see Theorem 1 below), the inequality 0< (c(s) -c{s))hP{s)ds < / (C(s) - c(s))/V(s) ds < oo together with non-oscillation of (2) ensure non-oscillation of (1). As an immediate consequence, we obtained a comparison theorem, where in the place of the equation which is being perturbed, we have the half-linear Euler equation W))' + ^{y) = o. (4) Here, 7p = is the so-called oscillation constant of (4), since it is the greatest possible constant for which the Euler equation is non-oscillatory for larger constants at that place the equation oscillates. Its principal solution is known exactly and is equal to £-1 t v . Another well-known equation that lies on the boundary between oscillation and non-oscillation is the so-called (generalized) Riemann-Weber equation (also referred to as the Euler-Weber equation or just the Euler-type equation). However, the principal solution of this equation cannot be expressed explicitly and only its asymptotic form is known; hence, the criterion from [1] cannot be applied to it. This was the reason for mentioning the open problem in [1], whether the principal solution in the criterion can be replaced by a function, which is, in some sense, only close to it. As the technique concerning the so-called modified Riccati equation has been developed in more depth over the last few years (see, for example, [10]), we can now show that the answer is positive. The paper is organized as follows. In the next section, we recall the Riccati technique, including the usage of the modified Riccati equation, the concept of the principal solution, technical lemmas, and remind the original theorem from [1] in its full version. In the section with the main results, we state and prove the main theorem and show some of its consequences for Riemann-Weber-type equations. The last part brings several concluding remarks. Mathematics 2021, 9,502 3 of 10 2. Preliminaries Supposing that Equation (1) is non-oscillatory, it is a well-known fact that if x is its solution, then the function w = r s°lves the- relevant Riccati equation zv'{t) + c{t) + {p-iy-V{t)\iv{t)\v = 0, l=yzi' (5) on some interval of the form [T, oo), and conversely, the solvability of (5) on an interval [T, oo) guarantees non-oscillation of (1). Here, we refer to the basic literature, for example, [8] (Section 1.1.4), for introduction to the theory (see also [11]). It can be shown (as introduced by [12]) that among all non-oscillatory solutions of (5), there exists the minimal one w, for which any other solution of (5) satisfies the inequality w(t) > w(t) for large t. Then, the solution of (5) given by x = Kexpj ^ r1"^(s)cl>-1(K>(s)) rfsj, is called "principal" and it is related to the minimal solution of (5) by the formula w = r(x'/x). Note that O-1 is the inverse operator to O and q is the so-called conjugate number to p, and ^ + ^ = 1 holds. The concept of the minimal solution of the Riccati equation is also known from the theory of linear differential equations, where the so-called integral characterization holds. Its possible extension to half-linear equations was studied, for example, in [13,14]. In [14], it was shown that the condition J°° r^x2^x/^p-2 = 00 is under certain assumptions necessary or sufficient for x to be the principal solution, but a complete "both-way" integral characterization has not been proven. Now, let us turn our attention to the modified Riccati technique. Let h(t) be a differen-tiable function such that h(t) > 0 and h'(t) ^ 0 for large t, and let us use the notation R(t) := r(t)h2(t)\h'(t)\p-2, G(t) := r(t)h(t)®(ti (t)). It was shown, for example, in [10] (Lemma 4) that a neigborhood of infinity solvability of (5) (and hence, also non-oscillation of (1)) is equivalent to solvability of the so-called modified Riccati equation v'{t) + h{t)l[h]{t) + (p - l)r1^(f)^^(f)W(^,G) = 0, (6) where H(v,G) := \v + G\q -q-x(G)v- \G\q. The solution v of the modified Riccati Equation (6) and the solution w of the Riccati Equation (5) satisfy the relation v = hpw — G. The behavior of H(v, G) was deeply described, for example, in [10], and we present here only those parts of its Lemma 5 and 6, which are relevant for us. Lemma 1. The function H(v, G) has the following properties: (i) H(v, G) > 0 with the equality if and only ifv = 0. (ii) For every L > 0, there exist constants K\ = K\{L) > 0,k2 = Ki{F) > 0 such that Ki|G(f)|,_2p2 < H(v,G) < K2\G{t)\q-2v2 for any t and v satisfying | ^ | < L. The nonnegativity of solutions of the modified Riccati Equation (6) was studied in several papers. The following lemma summarizes results which are already adjusted to Mathematics 2021, 9,502 4 of 10 our needs and based on Lemma 4 and a part of the proof of Theorem 4 in [15] (for more resources see references therein). Lemma 2. Let h be a positive, continuously differentiable function, such that h'{t) ^ 0 and h(t)l\h\(t) >0 for large t. Let j'°° JT1 (t) dt = oo and (liminf \G(t)\ > 0 and limsup\G(t)\ < oo) or lim |G(£)| = oo. t^co t^co Then, all proper solutions of (6) are nonnegative. Finally, let us present the main theorem of [1]. Note that h is here the principal solution, and assumption (7) is the condition appearing in its possible integral characterization. Theorem 1. Let J°° r1_?(f) dt = oo. Suppose that Equation (3) is non-oscillatory and possesses a positive principal solution h, such that there exists a finite limit and lim G(t) =: L > 0 /CO R~1(t)dt = oo. (7) Further, suppose that 0 < C(s) ds < oo and f-CO f-CO 0< {c{s)-c{s))hP{s)ds < (C(s)-c(s))^(s)ds < oo, all for large t. If Equation (2) is non-oscillatory, then (1) zs also non-oscillatory. 3. Main Results In this section, we present the main theorem and its corollaries. Theorem 2. Suppose that there exists a positive continuously differentiable function h{t) such that h'{t) 7^ Ofor large t and the following conditions hold: /CO R-1(s)ds = oo, (8) h(t)L[h](t) >0, (9) h(s)L[h](s) ds < oo, (10) (liminf |G(t)\ > 0 and limsup\G(t)\ < oo) or lim |G(f)| = oo, (11) t^co t^co all for large t. Let the inequality h(s)L[h](s) ds < J (c(s) - c(s))/V(s) ds < J (C(s) - c(s))ftP(s) ds < oo (12) be satisfied. Then, if Equation (2) zs non-oscillatory, Equation (1) zs non-oscillatory too. Proof. Suppose that Equation (2) is non-oscillatory. Let x be its solution. Then, the function w = rO ( y j solves on an interval [T, oo) the relevant Riccati equation w '{t) + C{t) + {p-l)r1-i{t)\w{t)\i =0 Mathematics 2021, 9,502 5 of 10 and the function v = Ww — G solves the modified Riccati equation v'(t) + h(t)L[h](t) + (p - ly-iWh-ityHfr G) = 0. (13) Because H(v, G) > 0 (see Lemma 1) and h(t)L[h] (t) > 0 for large t by (9), we observe that v'(t) < 0 and the function v(t) is non-increasing for large t. According to Lemma 2, the function v(t) is non-negative, and there exists a non-negative finite limit lim^oo v(t). If lim^oo | G(t) | = oo, then we immediately see that | ^ | —>■ 0 for t —>■ oo. Now we show the same for the remaining case if liminff_>oo |G(f)| > 0 and limsup^^ \G(t) | < oo. Integrating (13) over the interval [T, t] yields v(T) - v(t) = j\(s)L[h](s) + (p - l)r1-^(s)h-\s)H(v(s),G(s)) ds, and hence, v(T) > j\(s)L[h](s) + (p-l)r1-^(s)h-'^(s)H(v(s),G(s))ds. (14) Now we have (suppressing the argument) hL[h] = h{(r®(h'))' + c<$>(h) + (C - c)4>(ft)} = hl[h] + (C - c)W and thanks to (10) and (12), we observe that /°° h(t)L[h](t) dt < oo. Let t -> oo in (14) imply the convergence of the integral (p_ l)r1-?(f)/z-?(f)H(^(f),G(f))df < oo. With respect to our assumption—the first part of (11)—there exists a constant L > 0 and T\ > T such that | | < L for t >T\. By Lemma 1, there exists K > 0 such that K|G(f)|?"V(f) < H(v(t),G{t)) for £ > Tlr which means K^-rH(f)r'WH(,,W'GW) fOT f-Tl' Integrate the inequality over the interval [T2, oo), where T2 > Ti: /■oo -,2 /a a /-oo ^2 J YJjjdt< J r1~''(t)h-'>(t)H(v(t),G(t))dt. By (8), we see that v —> 0 for £ —> 00, and hence, v satisfies the integral equation h{s)L[h]{s) + {p- l)r1-c'{s)h-c'{s)H{v{s),G{s))ds. (15) Now let us define the following integral operator f-CO F{u)= / h{s)l[h]{s) + {p-l)r1-c'{s)h-c'{s)H{u{s),G{s))ds on the set U = {u(t),0 < u(t) < v(t),t 6 [T,oo)}. Our aim is to show that F on Lf fulfills such conditions that it has got a fixed point. Up to this point, first observe that H'u(u,G) = q^-\\u + G|) - qQ-^G), Mathematics 2021, 9,502 6 of 10 and since O-1 is increasing and u > 0 on U, it means that > 0, that is, H is increasing in the first variable. Let us take functions U\(t), u2{t) such that 0 < U\(t) < u2{t) < v{t)> then the inequality F(uj) < F(u2) holds too. To verify that the operator F maps the set IT to itself, we consider the inequality 0 < F(0) < F(u) < F(v) < v. (16) The middle two inequalities hold on IT according to the previous paragraph. Since F(0) h(s)l[h](s) ds h{t)L[h]{s) + (c(s) - £r(s))ftP(s) ds, the first inequality in (16) holds by the first inequality in (12). The last inequality /CO h(s)l[h](s) + {p- l)r1-c'{s)h-c'{s)H{v{s),G{s))ds < v follows from the fact that hl[h] = hl[h] + {c- c)W, hL[h] = hl[h] + (C - c)W together with (12) and (15). Furthermore, F(U) is obviously bounded on closed subintervals of [T, oo). To show that F is uniformly continuous, let u 6 IT be arbitrary, e > 0 and take t\,t2 G [T, oo] such that (without loss of generality)^ < t2. Denote/(f) = h(t)l[h](t) + (p - l)r1^(f)/z^(f)H(w(f), G(t)). We have \F(u)(t2) - F(u)(h)\ < f(t)dt h(t)l[h](t) dt f(t)dt + f(t)dt (p - l)r1~''(t)h-''(t)H(u(t),G{t)) dt Since both the integrals converge, there exists 5 such that each of the integrals in absolute value is less than | for 112 — t\ \ < 5 and \F(u)(t2)-F(u)(tl)\ <| + |=e- Hence, F is uniformly continuous. Using the Schauder-Tychonov theorem, there exists a fixed point of F on Lf such that F(u) = u and u solves the integral equation u(t) h(s)l[h](s) + (p- l)r1-\s)h-^(s)H(u(s),G(s)) ds and also the modified Riccati Equation (6), and w = h p(u + G) is a solution of the Riccati equation joined with (1). Hence, Equation (1) is non-oscillatory. □ As an immediate consequence of the previous theorem, we have the following statement. Corollary 1. Let the assumptions of Theorem 2 be satisfied. Then, the oscillation of Equation (1) implies that of (2). Now, for the sake of clarity, recall that by log we mean the natural logarithm, logfc stands for an iterative logarithm, and Log • is a product of these functions according to the following definition: logj t = log t, logk t = logfc_1 (log t), k>2, Logjt = n[=1 logk t. Mathematics 2021, 9,502 7 of 10 Let us consider the generalized Riemann-Weber half-linear equation with critical coefficients LRW[x] := (*(*'))'+ I 2£ + £ up tP k>(x) = o, (17) p—1 where heN and \ip = \ y—p— J • "^h6 consequence of Theorem 2 for the case where the non-oscillatory Equation (3), which is being perturbed, is set to be the Equation (17), reads as follows. Corollary 2. Suppose that the condition L holds for large t. If the inequality p-i I t v Log?t > 0 (18) fds< I (c - cjsP-1 Log n(s) ds < I (C-Č)SP-L Log n (s) ds < oo, where f(s) is defined by (19) (see below) and c(t) = {-jf + YJf=\ fP ^ it )>z's satisfied, and if Equation (2) is non-oscillatory, then Equation (1) is non-oscillatory too. Proof. First, note that Equation (17) is non-oscillatory, and it has, in a certain sense, the largest possible coefficient function c, for which the non-oscillation is preserved. Indeed, equation Fp + ;tl *pLog?i tPLogznt |cD(x) =0 is conditionally oscillatory, \i = \ip is its oscillation constant, and it is oscillatory for \i> \ip and non-oscillatory for \i < \ip. The asymptotic formulas for the two linearly independent non-oscillatory solutions of (17) were derived in [16]. These solutions are asymptotically equivalent to the functions h(t) = fV Loglt, h2(t) = fV Loglt\o%l+11 and h\ is asymptotically close to the principal solution. Let us take h(t) = h\(t) in Theorem 2 and check the conditions. p-i l Došlý in [17] showed that for h(t) = t v Log^t and the operator defined in (17), we have h'(t) = ^—^-rrLogitl 1 + f--i-- I and f(t) := h(t)L RWl (t) = t log31 27pp(2- 3(p-l)2 + o(l) as t —>■ oo. (19) Thus, (10) holds (the calculation can be found in [17] above the relation (3.9)). The condition in (9) is reduced to (18). Next, as t —>■ oo, we have R(t)=h2(t)\h'(t)\P-2 1 2-p tLogn(t)(l + o(l)) Mathematics 2021, 9,502 8 of 10 and jtR-\s)ds=(^>j2 Plog„+1(f)(l + o(l)) which is divergent for t —»■ oo, so (8) holds. Further, G(i) = h(t)d>(h>) = Logn(t)(l+o(l)) and it tends to infinity for t —»■ oo, and hence, (11) is also satisfied. □ In the next corollary, we apply the results to the generalized Riemann-Weber equation with n + 1 terms in the sum as the testing Equation (2) in order to obtain a Hille-Wintner-type comparison criterion for the perturbed Riemann-Weber-type equation (*(*'))' + (7? + E + gi*)) = °- (2°) \tP j=1tPLoSjt J Corollary 3. Let the inequality f(s)ds< / g(s)sP-1Logn(s)ds< Hp loSn+l(f)' where f{t) is given by (19), hold. Then, Equation (20) is non-oscillatory. Proof. We take (20) in place of (1), p-i i in place of (2), (17) in place of (3), and h = h\ = t v Log vn t. Observe that Hp (c-c)=g(t), [C-c) = - 2 and tPL°gn+lW tL0gn(t)logZn+1(t) The integral \c - c)hv = r-*L_.-ds = -—^—-, Jt s Log„(s)log2+1(s) log„+1(f) as can be shown by the substitution logn+1(s) = u and with the use of the fact that Finally, let us verify the condition (9). We have hL[h] = hLRW[h] + (C - c)b? flog3f 27PP(2-P) 3(P-1)2 + 1 ' Hp tLogn(t)logzn+1(t) Mathematics 2021, 9,502 9 of 10 as t —>■ oo. Show that Lo8^ = o(-—^—,——) as £ —>■ oo: Hog3f KtLogn(t)log2n+1(t)J L°gnt lim ^ = lim L0*h{t) = lim 1*M> < lim ^« ^ " where e = ^. This limit tends to 0 as t —»■ oo since lim^oo = 0 for e > 0 (as can be shown by the L'Hospital's rule). Hence, hL[h] > 0 for large t. □ 4. Concluding Remarks (a) Let us mention that Corollary 3, as the specific application of Theorem 2 to concrete Equations (20) and (17), and the generalized Riemann-Weber equation with n + 1 terms, brings a result which is in compliance with the Hille-Nehari-type criterion, that was proved in [17] (more on Hille-Nehari-type criteria for (20) can be found also in [18]). Its non-oscillatory part says the following. Suppose that the integral J°° c{t)tp~^ Log tdt is convergent. If r°° .. limsuplogn+11 / c(s)sp lLognsds -3upr then (20) is non-oscillatory. (b) Let us observe that Theorem 2 can be applied also to the situation where the Euler £-1 Equation (4) is in the position of (3). We can use the exact principal solution h = t v for which m = (^)2 f R-\s)ds = {i+0{i))(^>j2 piog(o, G(0 = (^y^) (1 + 0(1)) and h(t)LE[h](t) =0 as t —>■ oo. Such a corollary was already presented in [1]. (c) Note that the perturbation (c — c) does not have to be less than (C — c) pointwise (then the Sturm comparison theorem would be sufficient) and c can oscillate around c as long as the integral inequality (12) holds. For results for Riemann-Weber-type half-linear equations with sums of periodic functions instead of constants, see [19]. (d) Finally, comment on the differences between Theorems 1 and 2. Firstly, in Theorem 2, we do not suppose J°° r1_? = oo anywhere. Next, the main difference is the fact that h is once a principal solution of (3) and once a function which is only close to that principal solution. The condition (8) is in both the theorems, and it is connected with the closeness of functions h to the principal solution. The condition (9), that is, hL[h] > 0 for large t, does not have its counterpart in Theorem 1, and here we have another difference between the theorems. The reason for this condition is in usage of Lemma 2. The assumption (10) is a variant on the condition 0 < J°° C(t) dt < oo from Theorem 1. The condition (11) is in fact an extension of the assumption of the existence of a finite limit lim^oo \ G(t) |. We might ask whether (11) could be replaced just by liminff^oo |G(f)| > 0, but certainly, it can be replaced by a weaker condition of the existence of the limit such that lim^oo \ G(t) \ > 0. Note that the first part of (11) holds for the case where Equation (3) is the Euler Equation (4), whereas the second part holds for the case of Riemann-Weber type Equation (17). The last difference is in (12) in the very first inequality. In Theorem 1, the integral Jf°° h(s)L[h] (s) ds is equal to 0 trivially, because h is an exact solution of (3). Mathematics 2021, 9,502 10 of 10 Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Conflicts of Interest: The author declares no conflict of interest. References 1. Došlý, O.; Pátíková, Z. Hille-Wintner-type comparison criteria for half-linear, second-order differential equations. Arch. Math. 2006, 42,185-194. 2. Došlá, Z.; Hasil, P.; Matucci, S.; Veselý, M. Euler type linear and half-linear differential equations and their non-oscillation in the critical oscillation case. /. Inequal. Appl. 2019,189,1-30. [CrossRef] 3. Fujimoto, K. Power comparison theorems for oscillation problems for second-order differential equations with p(t)-Laplacian. Acta Math. Hungar. 2020, 262, 333-344. [CrossRef] 4. Hasil, P.; Jaroš, J.; Veselý, M. Riccati technique and oscillation constant for modified euler type half-linear equations. Publ. Math. Debrecen 2020, 97,117-147. [CrossRef] 5. Sišoláková, J. Non-oscillation of linear and half-linear differential equations with unbounded coefficients. Math. Methods Appl. Sci. 2021, 44,1285-1297. [CrossRef] 6. Takaši, K.; Manojlovič, J.V. Asymptotic behavior of solutions of half-linear differential equations and generalized Karamata functions. Georgian Math. }. 2021, 28,1-26. 7. Kusano, T.; Yosida, N. non-oscillation theorems for a class of quasilinear differential equations of second-order. Acta Math. Hungar. 1997, 76, 81-89. [CrossRef] 8. Došlý, O.; Řehák, P. Half-Linear Differential Equations. In North Holland Mathematics Studies 202; Elsevier: Amsterdam, The Netherlands, 2005. 9. Kusano, T; Yosida, N.; Ogata, A. Strong oscillation and non-oscillation of quasilinear differential equations of second-order. Differ. Equ. Dyn. Syst. 1994,2,1-10. 10. Došlý, O.; Fišnarová, S. Half-linear oscillation criteria: Perturbation in term involving derivative. Nonlinear Anal. 2010, 73, 3756-3766. [CrossRef] 11. Agarval, R.P; Grace, S.R.; O'Regan, D. Oscillation Theory for Second-Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations; Kluwer Academic Publishers: Dordrecht, The Netherlands; Boston, MA, USA; London, UK, 2002. 12. Mirzov, J.D. Principal and nonprincipal solutions of a non-oscillatory system. Tbiliss. Gos. Univ. Inst. Prikl. Mat. Trudy 1988, 31, 100-117. 13. Fišnarová, S.; Mařík, R. Half-linear ODE and modified Riccati equation: Comparison theorems, integral characterization of principal solution. Nonlinear Anal. 2011, 74, 6427-6433. [CrossRef] 14. Došlý, O.; Elbert, A. Integral characterization of the principal solution of half-linear, second-order differential equations. Studia Sci. Math. Hungar. 2000, 36,455-469. 15. Došlý, O.; Fišnarová, S.; Mařík, R. Power comparison theorems in half-linear oscillation theory. /. Math. Anal. 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[CrossRef] mathematics ÍMDPIJ Article Use of the Modified Riccati Technique for Neutral Half-Linear Differential Equations Zuzana Pátíková *'*© and Simona Fišnarová 2© 1 Department of Mathematics, Tomas Bata University in Zlín, Nad Stráněmi 4511, 760 05 Zlín, Czech Republic 2 Department of Mathematics, Mendel University in Brno, Zemědělská 1, 613 00 Brno, Czech Republic; fisnarov@mendelu.cz * Correspondence: patikova@utb.cz; Tel.: +420-57-603-5005 Abstract: We study the second-order neutral half-linear differential equation and formulate new oscillation criteria for this equation, which are obtained through the use of the modified Riccati technique. In the first statement, the oscillation of the equation is ensured by the divergence of a certain integral. The second one provides the condition of the oscillation in the case where the relevant integral converges, and it can be seen as a Hille-Nehari-type criterion. The use of the results is shown in several examples, in which the Euler-type equation and its perturbations are considered. Keywords: half-linear neutral differential equation; oscillation criteria; modified Riccati technique MSC: 34K11; 34C10 check for updates Citation: Patikova, Z.; Fisnarova, S. Use of the Modified Riccati Technique for Neutral Half-Linear Differential Equations. Mathematics 2021, 9,235. https://doi.org/10.3390/math9030235 Academic Editor: Jozef Dzurina Received: 28 December 2020 Accepted: 20 January 2021 Published: 25 January 2021 Publisher's Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. [(g) ® Copyright: ©2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1. Introduction In this paper, we study the oscillatory properties of the second-order half-linear neutral differential equation (r(t)(z'(t)))' + c(í)<Í>(x(t(í))) = 0, z(t) = x(t) + b(t)x(a(t)), (1) where t > t0 and O(x) = \x\p~zx, pel, and p > 1. We suppose that the coefficients of the equation satisfy the usual conditions: r 6 C([to,oo),b 6 C1 ([to,oo),R^), c 6 C( [to, oo), R+), c is not identically equal to zero in any neighborhood of infinity, and b(t) < 1. (2) Concerning the deviating arguments, we assume that t, a £ C1([fo/00)/^)/ lim T(0 = 00 / t—>oo lim o~(t) = oo and We also suppose that t' > 0, t(í) < t, a(t) < t. /CO r1^ (t) dt = oo, (3) (4) where q denotes the conjugate number of p, i.e., q = j^ry, and the symbol J°° means that it does not matter what the lower limit of the integral is if it is large enough, and that the limit process is applied on the upper limit of the integral as it tends to infinity. The above setting and conditions (2)-(4) are intended to hold throughout this whole paper and in all of its statements. A differential equation is called neutral if it contains the highest-order derivative of an unknown function both with and without delay. This means that the rate of growth depends on the current state and the state in the past, as well as on the rate of change in the past, which enables a suitable description for many real processes. For example, the process of Mathematics 2021, 9, 235. https://doi.org/10.3390/math9030235 https: / /www.mdpi.com/journal/mathematics Mathematics 2021, 9, 235 2 of 13 growth of a human population ([1]) or a population of Daphnia magna ([2]) can be modeled by neutral differential equations. Neutral Equation (1) is called half-linear, as its solution space is homogenous but not additive (it only has half of the linearity properties), and it can also be classified as Emden-Fowler equation. Neutral half-linear/Emden-Fowler equations arise in a variety of real-world problems, such as in the study of p-Laplace equations, non-Newtonian fluid theory, the turbulent flow of a polytrophic gas in a porous medium, and so forth (see, for example, [3-6]). In recent years, the qualitative theory of Equation (1) has attracted considerable attention, and it has been studied under condition (4), for example, in [7-11] (see also the references therein). For the case where the integral in (4) converges, let us refer to [12,13]. If b(t) = 0, then the studied equation becomes a delayed half-linear equation, and its oscillation results are provided, for example, in [14-20]. By a solution of (1), we mean a differentiable function x(t) that is eventually not identically equal to zero, such that r(t)<£>(z'(t)) is differentiable and (1) holds for t > tg. Equation (1) is said to be oscillatory if it does not have a solution that is eventually positive or negative. In this paper, we formulate new oscillation criteria for Equation (1). One of them can be classified as a Hille-Nehari type statement. Our results are based on the modification of the Riccati technique. Instead of the usual Riccati inequality, we use the so-called modified Riccati inequality. The modified Riccati technique has been used in the theory of ordinary half-linear differential equations of the form (r(f)$(i'(f)))' + c(t)$(.(f))=0, (5) and it has been revealed that it is a useful tool that can be regarded as a replacement of the missing half-linear version of the transformation formula known from the classical oscillation theory of linear equations. For the related results concerning this method, we refer to [21-23] and the references given therein. We point out that, within the same approach, Hille-Nehari-type criteria for (5) were last studied in [24]. In Ref. [18], the modified Riccati technique was extended and applied to half-linear differential equations with delay: (r(f)$(i'(f)))' + c(t)O(.(T(f)))=0. (6) Here, we show that the method can also be extended for neutral half-linear equations and used to derive some oscillation criteria for (1). This paper is organized as follows. In the next section, we introduce the modified Riccati technique and formulate some preliminary results. In Section 3, we present our main results, the oscillation criteria for (1), and in the last section, we apply the results to a perturbed equation of the Euler type. 2. Preliminaries We start with the properties of the eventually positive solutions of (1) that are ensured with condition (4). By the function -1, we mean the inverse function to O, i.e., _1(;t) = \x\q~zx. Lemma 1. Suppose that x{t) is a solution of (1) that is positive on [to/00)- Then, there exists T > t0 such that z(t) > 0, z'(t) > 0, (r(t)(z'(t)))' < 0, (7) ^m>^(j^-\ (8) and (r(t)^(z'(t)))' < -c(tMz(T(t))(l - b(r(t))} (9) for t > T. Mathematics 2021, 9, 235 3 of 13 Proof. Condition (7) is a well-known statement and its proof can be found, e.g., in [7] (Lemma 3). Because r(t)<£>(z'(t)) is non-increasing, we have r(t)(z'(t)) < r(T(t))®(z'(T(t))), which can be rearranged into (8). Now, we observe that x(c(t(£))) < z(c(t(£))) < z(t(£)) and, in view of (1), 0 =(r(t)d>(z>(t)))' + c(tMz(r(t)) - b(T(t))x(a(r(t)))] > (r{t)$>{z'{t)))' + c(f)4>[z(T(f))(l - b{T{t)))], which implies (9). □ Grace et al. showed in [10] that, under some additional assumptions, condition (8) can be strengthened. Similarly to in [25], they considered the sequence #>((l-b(T(f)))c(f), K(t):= f ^{s) ds, H(t) :=1Z(t) + ^— ft1Z(s)^(1Z(T(s)))Q(s)ds for t > i\, where i\ is large enough, Grace et al. proved the following lemma. Lemma 2 ([10], Lemma 4). Assume that t is strictly increasing, Equation (1) has a positive solution x{t) on [to,00), and the condition f Q{s)(ll{s))ds>Q (12) Jr(t) holds for some q > 0 and a t that is large enough. Then, Z'(T«) >q>-ifSn(Q)r(t) z'{t) ~ V r{T{t)) for every n and t that are large enough, where gn (q) is defined by (10). Now, let us turn our attention to the Riccati technique. By our assumptions, conditions (2)-(4) hold, and we suppose that Equation (1) has an eventually positive solution x(t). Take ^m = kW-™V (b) Z{T{t)) By a direct differentiation, we have w,[t) = WW®))' _ _ 1)r(ty(t)*(*W(T(0) [) O(z(t(0) [P i,r[t,T[t) \z(T(t))\r Mathematics 2021, 9, 235 4 of 13 which, with the use of (9), gives W'(t) < -C(£)0(1 " b(T(t))) - (p - iy^(t)T'(t)Z-^l\w(ty. Assuming that there exists a positive function/(f) (one possible choice is O-1 ^r(^(f)) by (8)) such that we obtain the Riccati-type inequality of the form W'{t) < -C(f)4>(l " b(T(t))) - (p - iy-^t)T'{t)f{t)\w{ty. (15) Next, we introduce the modified Riccati technique. Let h(t) be a positive differentiable function, and put G{t) = r{t)h{T{t))®(^j^y (16) Using the modified Riccati transformation v(t) =h?(T(t))w(t) -G(t), (17) we obtain the so-called modified Riccati inequality (18) that is derived in the next lemma. Lemma 3. Suppose that Equation (1) has an eventually positive solution x(t) and w is defined by (13). Let f be a positive function satisfying (14), let hbea positive differentiable function, and let G be defined by (16). Then, the function v{t), given by (17), satisfies the inequality v'(t) + C(t) + (p- iy-«(t)T'(t)h-«(T(t))f(t)H(v(t), G(t)) < 0, (18) where C{t) = h{T{t)) and rmi1^1) ) +c(t)d>(h(T(t)))d>(l-b(T(t))) (19) H(v,G) = |p + G|'-^_1(G)p- |G|'. (20) Proof. By a direct differentiation, we obtain v'(t) = phP-1(T(t))h'(T(t))iJ(t)w(t) + hp(r(t))w'(t) - G'(t) = ph'iTityy^h-^Tit))^) + G(t)) + w{T{t))w'{t) - G'(t), and with the use of (15), we have (suppressing the argument t) v' < pti(T)T'h-\T)(v + G) -G' + hp{T) [-cO(1-&(t)) - (p-iy-V/M?] = ph'(T)r'h-1(T)(v + G)-G'-hp(T)c-iy-lT,h-l(T)f[\v + G\l-q®-1(G)v-\G\l], Mathematics 2021, 9, 235 5 of 13 where C = —ph'(T)T,h~1(T)(v+ G) + G + hP{T)c{l - fe(T)) + (p - l)r1-tT,h-t(T)f\q<$>-1(G)v+ |G|' = ftP(r)c4>(l - fe(r)) + G' - pfc'(t)t'fc_1(T> - pti(t)t'Tr1 (r)rft(t)4> / |ft'(r)|P + pr^^'h^ (^fr^h^^^—^v + (p - l^V/r^t)/VW(t) = ftP(r)c4>(l - fe(-r)) + G' - rlft'^IPr'/1-?. Since ^M^yft(T(f))+r(f)|ft'(T(f))|pT'(f)/1-P(^ we have C(f) = ftP(T(f))c(f)*(l " fc(T(f))) + ( 'ft'(T(f)) ft'(T(f)) ft(T(f)) = h(T(t)) r(t)$> f(t) fit) + cm(h(T(t)))^(l-b(T(t))) Hence, C(t) = C(t) and the lemma is proved. □ Similarly to in [18], we have the following two statements. In the first one, we formulate estimates for the function H(v, G) from (20). Note that by applying these estimates in (18), we obtain an inequality that is, in fact, the Riccati inequality associated with a certain ordinary linear equation. The second statement gives sufficient conditions for the eventual non-negativity of the solutions to (18). By studying the proof of the original statement in [18], one can easily see that it also holds for the neutral version of the modified Riccati inequality (18). Lemma 4 ([21], Lemma 5 and Lemma 6). The function H(v, G) defined by (20) is non-negative and H(v, G) = 0 if and only ifv = 0. Furthermore, if lim inf | G(t) | > 0 and v{t) —>■ Ofor t —>■ oo, then 0, there exists a constant K > 0 such that H{v{t),G{t)) > K\G{t)\^zvz{t) for any t and v satisfying \ v(t)/G(t)\ < T. Lemma 5 ([18], Lemma 2.5). Let hbea positive continuously differentiate function such that h' 7^ Ofor large t and C(t) > Ofor large t. Moreover, let either limsup \G(t)\ < oo and or f(tV(t) r?-1(f)W(T(f)) dt = oo lim \G(t) \ = oo (21) (22) (23) Mathematics 2021, 9, 235 6 of 13 and ^(f(t))T'(t) r(t)h2(T(t))\h'(T(t))\P-2 tg such that (1) has a solution x(t) that is positive for t 6 [T, oo), and condition (14) holds on this interval. Then, v(t) defined by (17) satisfies (18), and hence, v'(t) < -C(t) -(p- iy-«(t)T>(t)h-«(T(t))f(t)H(v(t),G(t)). Integrating the inequality from t\ > T to t, we get v(t) Ofor large t. Moreover, let either f00 Tit) limsup |Gi (01 < 00 and I -1, ——-— dt = oo U 71 J r«-\T(t))h«(T(t)) or r°° _1 lim |Gi(f)| = oo and / R, (f) dt = oo. (27) Mathematics 2021, 9, 235 7 of 13 /CO C1{t)dt = oo, (28) f/zen Equation (1) is oscillatory. The second and last theorem is of the Hille-Nehari type and concerns the case where the integral in (25) is convergent. We present a version with the general function /, and G, C, and R are given by (16), (19), and (26); however, one can also formulate the special case of the theorem with the function f(t) = O-1 (-frrrr) and G\, C\, and R\, similarly FRF)) to in Corollary 1. Recall that the same types of results for half-linear Equation (5) were proved in [26], and for delayed half-linear Equation (6), comparison theorems providing qualitatively similar results were presented in [18]. Theorem 2. Let hbea positive continuously dijferentiable function such that h' ^ Ofor large t and let f be a positive function such that C(t) > Ofor large t, f°° R_1(£) At = oo, liminf G(t) > 0, anA limsupG(f) < oo or lim G(t) = oo. (29) Suppose that J°° C(t) At < oo. If ft 1 f°° 1 liminf / R~1(s)ds C(s) ds > —, (30) t^co J Jt 2q then Equation (1) is either oscillatory or in every neighborhood of oo, there exists t* such that z'{^)f^t*) ^ 1 for all solutions of (1). Proof. Suppose, by a contradiction, that there exists T > tg such that (1) has a solution x(t) that is positive for t 6 [T, oo), and condition (14) holds on this interval. All conditions of Lemma 5 are satisfied. Indeed, conditions (21) and (23) are given in (29), condition (24) is, in fact, J™ R^^dt = oo,and (22) can be written in the form /°°l?-1(r)|G(r)|2-'7 dt = oo;this follows from (24) and the fact that 0 < liminf \G{t)\ < lim sup \ G(t)\ < oo. With respect to Lemma 5, the function v(t) defined by (17) is eventually non-negative. We show that lim v(t) = 0. It follows from (18) that v'(t) < 0; hence, the limit exists and is non-negative and finite. Integrating (18) from Tj to t (Tj > T) yields 0(7i) - v(t) > f C(s) ds+(p- 1) f r1-^(s)T'(s)h-^T(s))f(s)H(v(s), G(s)) ds. Since v(t) > 0, we have > f C(s)ds + (p-l) f r1-^(s)T'(s)h-^T(s))f(s)H(v(s),G(s))ds. Both the integrals in the inequality are non-negative, and letting t —>■ oo, we see that the integral ^ r1^(s)T/(s)/i^(T(s))/(s)H(v(s),G(s))ds is convergent. With respect to conditions liminf G(t) > 0 and lim v(t) < oo, there exists a positive constant M and T2 > T\ such that | \ < M for t > T2. According to Lemma 4, there exists K > 0 such that K\G(t)\l-2v2(t) < H(v(t),G{t)) for t > T2. Mathematics 2021, 9, 235 8 of 13 Since R(t)\G(t)r2 = ^gf», we have (0/(0 < r^(t)T'(t)h-q(T(t))f(t)H(v(t),G(t)) fort > T2. Integrating the last inequality from T3 to t (T3 > T2) and letting £ —>■ 00, we obtain As J°° R_1(f) = 00, the last inequality implies that lim v{t) = 0. Now, we integrate (18) from t to 00 to obtain /CO f' CO C(s)ds + (p-l) J r1-c'{s)T'{s)h-c'{T{s))f{s)H{v{s),G{s))ds. Let e > 0. According to Lemma 4, there exists a T4 large enough such that HK0,G(0)>(fcl)-£)^)|G«r2 i(£^_eV?-2(f)Ä?-2(T(f))|Ä/(T(f))|2-P/P-2(f)p2(f) for t > T4. Hence, 2 J Jt R(s) for t > T4. Denoting e = (p — l)e, the inequality becomes v{t) > J™C{s)ds + (| -e) R^1{s)v2{s)ds and multiplication by 7?,(£) = Jt R^1 (s) ds gives > jf C(s) ds + (| - e)ft(i) |r^WsMs))2 ds- (31) With respect to (30), there exists 5 > 0 such that liminf TZ(t) /(°° C(s) ds > ^ + Furthermore, we observe that TZ(t) Jf°° ds = 1. There are two possible options: either liminf 1Z{t)v{t) < 00 or liminf 1Z{t)v{t) = 00. Let us discuss both cases. t—>co f—>co (a) Suppose that liminf TZ(t)v(t) := L < 00. From (31), one can see that L > 0. For every £ > 0, there exists a T5 large enough so that (TZ(t)v(t))2 > (1 — E)2L2 for t > T5. Estimations of the terms in (31) give (l + £)L>l + ^+(|-£)(l-£)2L2. Letting £, £ —>■ 0 gives a contradiction, since then, L> ^ + S+h2^q-(L--)2 + S<0. ~ 2q 2 2 V q ir from (31), it follows that (b) In the case where liminf 1Z{t)v(t) = 00, we denote mit) = fnf{7£(sWs)}. Then, Mathematics 2021, 9, 235 9 of 13 1l(t)v(t)>±- + S+^-š)m2(t). 2q Since the function m is nondecreasing, we have for s > t: K(s)v(s) > 1 + S + (I - e) m2(s) > 1 + í + (I - g) m2(t), and hence, m(ř) > ^ +S+ (| - ě)m2(ř) >(§-ě)m2(ř), which gives a contradiction with the assumption liminf lZ(t)v(t) = oo. The proof is finished. □ 4. Examples Consider the Euler-type equation + b0x(o-(t)yyy + ^o(x(aí)) = o, (32) where A 6 (0,1), a(t) < t, lim a(t) = oo, and b0 6 [0,1). Equation (32) is of the form (1), where r(r) = 1, c(r) = r(f) = At, and fc(r) = b0. £-1 Example 1. Ta/ce h(t) = t ? ; then, by a direct computation, we have Gi(t) = p—i , Ri(t) = p-2 (33) and T'(t) ■dt rí-1(T(ř))W(T(ř)) J t Hence, condition (27) is satisfied. Furthermore, we have dt Ci(f) = (Ar)"'1 £-1 (Aí) p = r = r i-P\ / p_i\p-i' (Ař) p +c(ř)4>(l-M (Ař) p 1 ^P\p~1tP~2 + c(t)(\ - M(Ař)p~2+p + c(ř)*(l -&0)Ap_1íp + 7^(1 - b^xp-1 (34) 77ze positivity of the expression Equation (32) is oscillatory if i\P 7 > + 7(1 — ^o)Ap 1 implies (28). So, by Corollary 1, IV 1 AP-!<1)(1 - fc0)' (35) 77zzs corresponds to the result known in the case where b0 = 0 (equations with delay, see [18]) and also with the case where bo = 0 and A = 1 (ordinary equations). Mathematics 2021, 9, 235 10 of 13 Example 2. Condition (35) can even be strengthened for a class of Equation (32), which satisfies (12) with q e (0, \). By a direct computation (or see [10]), one can show that for (32), Q = (i - boy-^AP-1 (i + ^-ap-Hi - fccF-1?)" 1 lo§(p4r) ■ Note ftof here and in what follows, the symbol log stands for the natural logarithm. If q e (0, |), according to Lemma 2, we can use in place of the positive function f (t) from (14) the function ^ 1 ifr(T(t))\ w^ere S(q) zs defined by (11). Then, functions R, g, and C become g2(t) = r(T(t))h(T(t)Mh'(r(t)))(g(Q))-\ C2(t) = h(T(t)) r{T{t)) ))})' + c{t){h{T{t))){l - b0) Mt) = KT(0)^yft2(T(0)|ft'(T(0)lp-2(^(c))-1- In our setting for Equation (32) and again with h(t) = t v , these read as (hit) p—i (g(Q))-\ Rt(t) p-2 (giQ))-1* and c2(t) = r1 (g(Q))-1 + 7^(i-b0)AP-1 Since (21) and (22) hold and (25) is implied by the positivity of the expression -IV, (g(Q))-i + 7^(l-b0)A Equation (32) is oscillatory, according to Theorem 1, provided that IV 1 p—l 7 > V ) (g(Q))AP-^(l-b0)-This corresponds to the condition derived in [10]. In the theory of ordinary equations, i.e., in the case where A = 1 and bo = 0 in (32), it is known that Equation (32) oscillates if and only if (35) holds. This means that the constant ,-l\P 7 = \ ~p~) *s tne critical constant between oscillation and non-oscillation of (32), and it is natural to study perturbations of the Euler-type equation with this critical constant and to find critical constants in the added terms. This corresponds to the concept of conditional oscillation (for details, see [23,27] and the references given therein). In the case of the delayed and neutral equations, there is not such a boundary between oscillation and non-oscillation of (32). However, based on the results known from the ordinary case, let us study the neutral version of the Euler-Weber-type equation (O(i(f)+fe0i(^)))')' + p_i\p p + AP-l<$>{l-bo)tP tP log2t 0(x(A(£)) = 0. (36) Mathematics 2021, 9, 235 11 of 13 Example 3. For (36), we use Theorem 2 with /(f) = O 1 (-frryr) • Similarly to in Example 1, j(T(t)) it>e tofce /*z(f) = f p , if e obtain (33), and, zuzf/z f/ze use of the relevant coefficient function c(f) in (34), we have Ci(t) = -^1-Mi-b0w-1 flog f and t Jt slogzs logf Since R^1 = (^~t) ^ j/ condition (30) becomes T) ""%(!-MA""1 >^ which, by Theorem 2, implies that Equation (36) is oscillatory if 1 /p_!\p-i ' (37) Nofe f/zaf in f/ze ordinary case where A = 1 and fro = 0, the constant from (37) zs critical, which means that (36) zs oscillatory if and only if u > \ {~jr)V ■ £-1 I Example 4. Let us consider the perturbation of'the Euler-Weber equation. Take h(t) = t v log? f and consider Equation (1) with r{t) = 1, r(f) = Af, and b{t) = 0. Observe that d — 1 _l I h'(t) = -f v log? f By a direct computation, we see that p—i l + 7 1 l)logf. Gi(0= (^) log(Af)(l + 0(l)), 1 p-2 R^={p) flog(Af)(l + 0(l)) as t —>■ oo. Wzf/i f/"ze use o/f/ie power expansion formula s(s — 1) 9 s(s — l)(s — 2) , , ,s (1 +x)s = 1 + sx + —--xz + —---Lxi + o{xi) as x -> 0, v ; 2 6 ^ one can s/toa; that for 1J AP-Hp 2\p-l) AP-HP\ogz(At) fPlog2(Af)log2(log(Af))' we have uAP-1 flog(Afyio^z(log(Af)) Because JR-\t)dt^(l-lY Plog(log(Af)) CiW^,, 2n ,,,„(l+o(l)) asf Mathematics 2021, 9, 235 12 of 13 and Ci{t)dt log(log(Ai))' condition (30) becomes 1 _ p-1 p —1 Since i = the considered equation is oscillatory if 1 /P-1 2\ p J AP-1 5. Conclusions The aim of this paper was to study how the modified Riccati technique can be applied to Equation (1), which has not been tried for neutral equations before, and what results this approach can provide. According to our results, the modified Riccati method is applicable to Equation (1), and it can be used to find new criteria, for whose proofs it is enough to manipulate the modified Riccati inequality. We have presented two new oscillation criteria and illustrated their uses in examples dealing with a half-linear Euler-type equation and its perturbations. Author Contributions: The authors contributed equally. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Conflicts of Interest: The authors declare no conflict of interest. References 1. Burton, T.A.; Purnaras, I.K. A unification theory of Krasnoselskii for differential equations. Nonlinear Anal. 2013, 89,121-133. [CrossRef] 2. Smith, EE. Population dynamics in Daphnia magna and a new model for population growth. Ecology 1963, 44, 651-663. [CrossRef] 3. Bohner, M.; Hassan, T.S.; Li, T. Fite-Hille-Wintner-type oscillation criteria for second-order half-linear dynamic equations with deviating arguments, lndag. 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[CrossRef] Hindawi Complexity Volume 2020, Article ID 2854574, 12 pages https://doi.org/10.1155/2020/2854574 Wiley Hindawi Research Article Differential Transform Algorithm for Functional Differential Equations with Time-Dependent Delays Josef Rebenda©1'2 and Zuzana Pátíková3 'CEITEC BUT, Brno University of Technology, Purkyňova 656/123, 61200 Brno, Czech Republic 2Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická 8, 616 00 Brno, Czech Republic 3Department of Mathematics, Faculty of Applied Informatics, Tomas Bata University in Zlín, Nad Stráněmi 4511, Zlín 76005, Czech Republic Correspondence should be addressed to Josef Rebenda; josef.rebenda@ceitec.vutbr.cz Received 23 August 2019; Accepted 31 December 2019; Published 28 February 2020 Guest Editor: Robert Hakl Copyright © 2020 Josef Rebenda and Zuzana Pátíková. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. An algorithm using the differential transformation which is convenient for finding numerical solutions to initial value problems for functional differential equations is proposed in this paper. We focus on retarded equations with delays which in general are functions of the independent variable. The delayed differential equation is turned into an ordinary differential equation using the method of steps. The ordinary differential equation is transformed into a recurrence relation in one variable using the differential transformation. Approximate solution has the form of a Taylor polynomial whose coefficients are determined by solving the recurrence relation. Practical implementation of the presented algorithm is demonstrated in an example of the initial value problem for a differential equation with nonlinear nonconstant delay. A two-dimensional neutral system of higher complexity with constant, nonconstant, and proportional delays has been chosen to show numerical performance of the algorithm. Results are compared against Matlab function DDENSD. 1. Introduction Functional differential equations (FDEs) are used to model processes and phenomena which depend on past values of the modelled entities. Indicatively, we mention models describing machine tool vibrations [1], predator-prey type models [2], and models used in economics [3]. Further models and details can be found for instance in [4, 5] or [6]. Differential transformation (DT), a semianalytical approach based on Taylor's theorem, has been proved to be efficient in solving a variety of initial value problems (IVPs), ranging from ordinary to functional, partial, and fractional differential equations [7-11]. However, there is no publication about systematic application of DT to IVP for differential equations with nonconstant delays which are functions of the independent variable. In this paper, we present an extension of DT to a class of IVPs for delayed differential equations with analytic right-hand side. Albeit the analyticity assumption seems to be quite restrictive, it is reasonable to develop theory for such class of equations [12, 13]. The paper is organised as follows. In Section 2, we define the subject of our study and briefly describe the methods we combine to solve the studied problem, including recalling necessary results of previous studies. Section 3 contains the main results of the paper, including algorithm description, new theorems, examples, and comparison of numerical results. In Section 4, we briefly summarise what has been done in the paper. 2 Complexity 2. Methods 2.1. Problem Statement. The problem studied in this paper is to find a solution on a given finite interval [tQ, T] C [0, oo) to an IVP for the following system of p functional differential equations of «-th order with multiple delays ax (t),..., ar (t): u(n) (f) = f(f, u(f),u' (f),...,u("-15 (f), ut («! (f)),. . •,ur (ar (f))), (1) where u(n) (f) = (u[n) (f),uf (t))T , u{k) (t) = (u[k) (f), . ..,««(£)), fc = 0, l,...,n- 1, and f = (f1,---,fp)T are ^-dimensional vector functions, u; («,(£)) = (u (a,- (f)), u' (a,- (f)),..., u(m'' (a,- (f))) are (m,- • p)-dimen-sional vector functions, mt rj0 > 0 for t € [tQ, T],i = 1,2,..., r, is in general a real function, that is, a time-dependent or time-varying delay. Constant and proportional delays are considered as special cases. In case that some at is a proportional delay, we do not require the condition rt (t) > rj0 > 0 to be valid at 0 if 10 = 0. Let t* = min1p (t))T must be prescribed on the interval [t\tQ]. DT algorithm for the case t* = t0 = 0 with all delays being proportional is described in [14]. DT algorithm for the case t* < t0 when all delays are constant is introduced in [15]. In this paper, we develop the algorithm for the case t* < t0 when at least one delay is nonconstant. To have a complete IVP, we consider system (1) together with initial conditions: u(£0) = v„, (2) u' (t0) = v1,...,u{n~1) (t0) = vn_1, and, since t* < f0, also subject to initial vector function O (t) on interval [t*,t0] such that O(t0) = u(t0),...,O^1)(t0) = u^1)(t0). (3) We consider the IVPs (l)-(3) under the following hypotheses: (HI) We assume that all the functions (f>j(t), j = \,...,p, are analytic in [t*,tQ], the functions at(£), i = 1,..., r, are analytic in [tQ, T] and the functions fj,j = \,...,p, are analytic in an open set containing [tQ, T] x [u(f0),u(T)]x...x[ur(ar(f0)), ur(ar(T))]. (H2) If a; (t) = qtt and mi = n in fj for some i € (1,..., r} and j € {l,..., p\, that is, jth equation is neutral with respect to the proportional delay ajy we assume that u\n) (a,- (£)) = OforZ € {l,... ,p), I # j. This hypothesis is included since if it is not fulfilled, the existence of unique solution of IVP could be violated. We note that these assumptions imply that the IVP (l)-(3) has a unique solution in the interval [f0,T]. 2.2. Method of Steps. The basic idea of our approach is to combine DT and the general method of steps. The method of steps enables us to replace the terms including delays with initial vector function O (f) and its derivatives. Then, the original IVP for the delayed or neutral system of differential equations is turned into IVP for a system of ordinary differential equations. For the sake of clarity, we include a simple explanatory example. Suppose that we have a system with three delays, one of each type considered: ax (t) = t - rx (£), a2 (t) = t - t2, and a3 (t) = q3t. We have to distinguish two cases: (a) If t0 = 0, applying the method of steps turns system (1) into u(n) (t) = f(f,u (£),..., u("~15 (£), V (4) 0: (f-t1(f)),2(f-t2),u3 (%t)), while (b) If t0 > 0, system (1) is simplified to u(n) (£) = f(f,u (£),..., u("~15 (£), (5) #! (f-t1(f)),02(f-t2),03 (q3t)), where ! (t - tx (t)) = (o(t- r, (t)), 3 (%t) = (O (q3t), O' (q3t),. . ., O M (q3t)), (6) and m; 1, t are the minimal roots with odd multiplicity of r equations: tp,(a-l)r+ti~ Tfi = tp-l,a' ^ = 1, • • • , f. (7) Complexity 3 If nonconstant nonproportional delays at appear in system (1), the principal set of discontinuity points is defined as follows: Definition 1. The principal discontinuity points for the solutions of system (1) are given by the set of points t , such that t01 = t0 and for p, a > 1, t are the minimal roots with odd multiplicity of r equations: Xyifp^o-Dr+t) p-i,ff> P= 1,. ■ > r. (8) Similar to the case of constant delays, we break the interval [t0,T] into subintervals f = I = l,...,K. We start with the mesh grid {f0,.. .,tK\ formed by the principal discontinuity points calculated using Definition 1. To improve convergence or performance of the algorithm, there is a possibility to refine the mesh grid by inserting other points into it. For more details on the principal discontinuity points and mesh grid, we refer to the monograph [16]. 2.3. Differential Transformation Definition 2. Differential transformation of a real function u(t) at a point t0 € R is 3){u(t)}[t0] = {U(k) [£0]}^0> where U(k) [tQ], k - th component of the differential transformation of the function u(t) at t0, k € N0, is defined as U(k)[t0] vdku(t) k\ dtk (9) provided that the original function u (t) is analytic in a neighbourhood of t0. Definition 3. Inverse differential transformation of {U (k) [£0]}/So is defined as co u (t) = 31-1 {{U (k) [t0] }~0} [t0] = £ U (fc) [t0] (t - t0)k. k=0 (10) In applications, the function u (t) is expressed by a finite i(t) = JjU(k)[t0](t-t0) (11) k=o As we can observe in (10), DT is based on Taylor series; hence, any theorem about convergence of Taylor series may be used. However, we would like to point out the paper [17] where the finest general explicit a priori error estimates are given. The following formulas are listed, e.g., in [18] and will be used in Section 3.3. Lemma 1. Assume that F(k)[t0] and U(k)[t0] are differential transformations of functions f (t) and u(t), respectively: If fit) = then F(k)[t0]= U(k + n)[t0]. If f(t)=tn, then F (k) [0] = S (k - n), where S(k - n) = Skn is the Kronecker delta. Iff(t) = e , thenF(k)[0] k\' (12) Remark 1. Similar formulas can be obtained using numerical approach called Functional Analytical Technique based on Operator Theory [19, 20]. The main disadvantage of many papers about DT is that there are almost no examples of equations with non-polynomial nonlinear terms containing unknown function u(t) like, for instance, f (u) = if I + u3 or f (u) = e"^^1. However, DT of components containing nonlinear terms can be obtained in a consistent way using the algorithm described in [21]. Theorem 1. Let g and f be real functions analytic near t0 and g(t0), respectively, and let h be the composition h(t) = (fog)(t) = f(g(t)). Denote 9>{g(t)}[t0] = {G(k)}Z0, 2>{f(t)}[g(t0)] = {F(k)}Z0, and 2>{(f°g) (t)}[to] = {H(k)}^l0 as the differential transformations of functions g,f, andh att0, g(t0), andt0, respectively. Then, the numbers H(k) in the sequence {H(k)}^i0 satisfy the relations H(0) = F(0) and k H(Jfc) = YJF(l)-Bk,i(G(l),...,G(k-l+ 1)), for k>l, (13) where Bkl {xx,..., xk_l+1) are the partial ordinary Bell polynomials. The following Lemma proved in [21] is useful when calculating partial ordinary Bell polynomials. Lemma 2. The partial ordinary Bell polynomials Bkl (x1;. ..,x,k_i+l), I = 1,2,. ..,k> I, satisfy the recurrence relation k-l+l ■ _ i Bk,l (Xi,..., Xk_M) = V —X;Bfc_;;_1 (Xj, . . . , Xfc_;_;+2), 1=1 K (14) where B00 = 1 and Bk0 = 0 for k>l. 3. Results and Discussion 3.1. Algorithm Description. Recall system (1) uw (f) = f(f, u (f), u' (f), • • • , u("- " (f), «!(«!(?)),...,«,(«,(?))), (15) with initial conditions 4 Complexity u(g=v0V(g = v1,...y"-1)(fo) = v„_1, (16) and initial vector function on interval [t*,t0] satisfying 0(fo) = u(fo),...,0<-1)(fo) = u^1'(fo (17) Further recall that in Section 2.2, we broke the interval [£0,T] into subintervals f = 1= l,...,K. Define h = [tm,t0]. Then, we are looking for a solution u(t) of the I VP (l)-(3) in the form u(t) uh(t), t€f, uh (t), t € I2, U; (f), t€lR, (18) where solution u7 in the jth interval f is obtained in the following way. We solve the following equation: u<;)(f) = /(f,u7.(f),u;.(f),...,u<;-1)(f), ujJ(a1(t)),...,ujir(ar(t))), (19) where (20) if «,(£) e J„for£ £ {1,...,;}, j e (1,... .K}. In case that «,(£) € 10 = [f*,f0] f°r t e then again Uj, (a,- (0) = ( (a,- (0), 0' («,■ M), ■ ■ ■, 0(mi) («; (0) (21) Application of DTat tj_x to equation (19) yields a system of recurrence algebraic equations: U7. (k + n) [thl\ = F(k, U7. (k), U7. (k + 1),..., U7. (fc + w - 1) (22) where the function F is the DT of the righthand side of equation (19) and involves application of Theorem 1. Next, we transform the initial conditions (2). Following Definition 2, we derive U7. (k) [tH] = ^i/W(f;-i)> for * = 0,1,.... n - 1, j € {1,.... 1C}. (23) Using (22) with (23) and then inverse transformation rule, we obtain approximate solution to (19) in the form of Taylor series: ^iit) = YvIik)[tj_l](t-tj_l)k, telj, (24) for all je {l,...,K}. To transform (20) correctly, we need the following theorem. Theorem 2. Let at (t) € ffort € f, where I € {l,..., j - l}. Let p € N. Denote 9){at (f)} [t^] = JA,- (k) [f^j}^. Then, S{uf(a,.(£))}[Vi] where B00 = 1, Bk0 = 0 for k>l, and 00 / X + v \ x=0\ x / (25) (26) for y>0. Proof. To prove (25) with p = 0, we use Theorem 1 with f(t) = w7( (£), g (t) = 0Cj (t), and h (t) = (f°g) (t). We immediately get k •Sfc,y(A(i)[Vi]"-"A^-y+i)[fM])> (27) for k>l. For A: = 0, Theorem 1 yields H(0) [fj_J = U7; (0) [at (ttl)] = Uh (0) [at (t^)] ■ \0 (A; (1) [f^J). Now, (25) for p > 0 is a consequence of Lemma 1 and it remains to prove (26). We recall that (28) k=0 As the assumption was that at {tj_x) € f, we may apply Definition 2 to (28) and obtain u,Sy)[«i(tH)] VuAt) dty (29) Substituting t0 = 0Cj {tand z = x + y gives (26). □ Complexity 5 3.2. New DT Formulas. In the applications, we also use the following DT formulas. Theorem 3. Assume that F(k)[t0] is the differential transformation of the function f(t) and r € R: (a) Iff(t) = f, then F(k) [t0] = [Y Wfc for all t such that\t-t0\<\t0\, W where ^ fc J = r (r - 1)... (r - fc + l)/k\ = (r)k/k\ and (r)k is the Pochhammer symbol. (b) If f(t) = \n(t), then F(k)[t0] = (-it1 / (k ■ tk0) for k> 1. 3.3. Applications. In this section, we introduce two test problems and show how the practical implementation of the presented algorithm looks like in concrete examples. Comparison of numerical results is given in Section 3.4. As the first test problem, we choose an IVP for a scalar equation with one nonconstant delay where the exact solution is known to be the exponential function e*. The purpose of including this example is to compare results obtained by DT against values of the exact solution and also against results obtained by Matlab function DDENSD which has been designed to approximate solutions to IVP for neutral delayed differential equations. Proof (a) Recall the Newton's generalisation of the binomial formula: if x and y are real numbers with \x\ > \y\, and r is any complex number, one has 00 / r \ r-k k x y , (30) where y^j = r(r - 1)... (r - k + \)lk\. Let us rewrite f as f = (t-t0 + t0)r = (t0 +(t- t0))r. Applying (30) yields tr=i(py0-k(t-t0)k k=0\lc (b) We start by proving the formula (31) (ln(f)) (k) (-1)^-1(^-1)1 tk (32) by induction. For k = 1, we have (ln(£))' = lit; hence, (32) is valid. Suppose that (32) holds for k. Then, (ln(f))(fc+1)=((ln(f))(fc))' /(-D^-DiY (-l)fc_1(fc- \)\(-k)fk~l (-If*! (33) -k+l Thus, formula (32) is valid for all k € N. Now by Definition 2, (-D k-l k ■ f{$ (34) □ Example 1. Consider the delayed equation: u (t)=u(t)-t + u(\n(t)), (35) with the initial condition u(l) = e, and with the initial function (f>(t) = e\ t € [0,1]. (36) (37) First we find the differential transform of the initial condition (36) which is U(0)[l]=e. Further denote ®K}[0] = (£(A:)[0]}^0 as the transformation of the exponential function with the center at 0 and 25{ln(£)}[l] = (F(fc)[l]}^0 as the transformation of the logarithmic function at 1, respectively. Then, Lemma 1 and Theorem 3 yield E(k)[0]=-, k\ \k-l (38) (-If F(k)[l]=-^-f-, forfc>l. 1 dfcln(f)" 1 [ (-Dfc 1 (fc - k\ dtk t-t ~V tk t=t0 For t € [l,e], equation (35) is transformed into (fc+ l)C7(fc+ 1) [1] = U(k)[l]-S(k)-S(k- 1)+H(fc)[l], (39) where k H(k)[l]=YJE(l)[0]Bk,i(F(l)[l],...,F(k-l+l)[l]), forfc> 1, H(0)[1] =E(0)[0] = 1. We have (40) 6 Complexity U(l)[l] = e- 1 -0+ 1 = e, H(l)[l] =£(1)[0].F(1)[1] = 1-1 = 1, 1/(0) [e] = u(e) = e, l/(l)[e] = 1/(0) [e] - e<5(0) - + H(0)[e] = e -e-0+e=e, [7(2)[1] =-([7(l)[l]-0-l+H(l)[l]) - (e- 1 + 1) = e--, 2 2 H(2)[l] = £(l)[0].F(2)[l]+£(2)[0]. (F(l)[l])2 [7(3)[1] =i([7(2)[l]+H(2)[l])=i.e.^ = e.i (41) Using the inverse transformation, we see that for t € [l,e], u(t)=e[ ! + (£-!)- a-i)2 a-i)3 3! t-i t e • e = e , (42) which corresponds to the exact solution to the IVPs (35)-(37). In the second step of the method of steps, i.e., in the interval t € [e,ee], we know that u(t) = e* for t € [l,e] and equation (35) is transformed into (k + \)U (k + \)[e\ = U(k)[e] - eS(k) - 8(k - 1) + H(k) [e], (43) where H(0)[e] =[7(0)[1], k H(k)[e]=JjU(l)[l]Bk,l(F(l)[e],...,F(k-l+l)[e]), i=i forfc> 1. (44) Here, F(k) [e], according to Theorem 3, are coefficients of Taylor series of logarithmic function with the center at e: F(0)[e] = 1, F(k)[e] (-D fc-i (45) k-ek -, forfc>l. Taking the values calculated in the first step and substituting them into the recurrence formulas (43) and (44), we obtain H(l)[e] =17(1)[1]-F(l)[e] =e-= 1, e l7(2)[e]=-(l7(l)[e]-0-l + H(l)[e]) = - \ee - 1 + 1] = -e\ H(2)[e] =l7(l)[l]-S2jl(F(l)[e],F(2)[e]) + l7(2)[l]-S2j2(F(l)[e]) = e-F(2)[e]+y (F(l)[e])2 =ei(^)+f-®2=0- 17 (3) [e] =i(C/(2)[e] + H(2)[e]) = iee, H(3)[e] =l7(l)[l].S3jl(F(l)[e],F(2)[e],F(3)[e]) + l7(2)[l]-S3j2(F(l)[e],F(2)[e]) + l7(3)[l] ■S3j3(F(l)[e]) = e • F(3) [e] +1 • 2 • F (1) [e] • F(2) [e] + ^(F(1) [e])3 2 1 1 (-1) lei 1 (I 1 1\ 2 6 e3 e2V3 2 6 U(4)[e] =](U(3)[e] + H(3)[e]) = -\ee. 4 4! Hence, for t € [e, ee], we have 1 ,1 , m (£) = ee + ee (t - e) + -e (t - e) + —e (t - e) (46) 1 4? '(t-e) el l + (t-e) (t - ef (t - ef (t - ef 3! 4! e t-e e - e (47) which again coincides with the exact solution to problems (35)-(37). In the second application, we have chosen an IVP for a nonlinear system of neutral delayed differential equations taken from the fully open access paper [18]. There are several reasons to test the proposed algorithm on the particular problem. The first is that the problem involves a nonlinear system of neutral equations of high complexity whose exact solution is unknown. Secondly, the proposed algorithm is a Complexity 7 complete differential transform version of the algorithm presented in [18] where modified Adomian formula has been used. Furthermore, the calculations done in [18] are shown only for the first step of the method of steps up to the first principal discontinuity point, whereas we continue calculations beyond that point in this paper. Last but not least, we want to verify performance and reproduce values obtained by DT and published in [18]. Rebenda et al. [18] has been submitted 4 years ago for the first time, and since that time, the Maple source code has been lost. Example 2. Consider a nonlinear system of neutral delayed differential equations: '1"(č-2)Ml(0 + ^(Ml(č))2 + "2( (£)+"2(f-l)"lQ (48) (49) m _ 1 «,(t with initial functions 0i (t) = e, 4>2(t)=t2, for t € [-2,0], and initial conditions «i(0) = 1, u'2(0) = 1, u"(0) = 1, u2 (0) = 0, »4(o) = o, u2 (0) = 2. For t € [0, fj, where tx ~ 0, 351734 is the minimal root off - (l/2)e~f = 0, using the method of steps, we obtain u[ = e{h(t)}[0] = {H1 (fc) [0] }^0, and we apply Theorem 1. First, applying DT to system (2) at t0 = 0, we get the recurrent system: (fc + 1) (fc + 2) (fc + 3) I/, (fc + 3) [0] + 28(k) (fc+l)(fc + 2)(fc + 3)(l-^r) U2(fc + 3)[0] = iQk "u^k-im-lu^km. Denote g(t)=t2/3; then, h(t) = (goUl)(t), and following Theorem 1, we obtain (-If fc! ' (52) (53) HiiOMO] =G1(0)[1]> k Hl (fc) [0] = £ Gl (I) [1]BW (Ul (1) [0],..., Ul (fc - I + 1) [0]), (54) forfc>l, where9){g(t)}[l] = {G1 (fc) [1]}^0 and, Theorem 3 ( 2/3 \ being applied, G^fc^l] = I ^ j for fc>0. Furthermore, the transformed initial conditions are U, (0)[0] = 1, tMi)[o] = 1, U1 (2) [0] = -, 1 2 (55) E72(0)[0] =0, U2(1)[0] =0, U2(2)[0] = 1. Using them, we compute the first three coefficients of the nonlinear term h(t): Hi(0)[0] =G1(0)[1] = 1, H, (1) [0] = G1 (1) [ 1] • Bu (U, (1) [0]) = ^ • 1 = I H1(2)[0]=G1(l)[l]-B2il(U1(l) [0], l/i (2) [0]) + G, (2)[1]-B„(t71(l)[0]) =|.|-|-1 =|. (56) Solving recurrent systems (52) and (53), we get fc = 0: IM3)[0] =i(e-2l/1(0)[0]+H1(0)[0] + l) 2 + e-2 fc = 1 U2(3)[0] =^(~2Ul (0)[0]) = -| : l/!(4)[0] =^(i[/1(l)[0]+U1(0)[0]) 1 / , w , \ 4e~2 +5 + _(H1(1)[0] + 1)=-^ 2 27' U2(4)[0] =1(2L71(0)[0]-^171(1)[0]) :2: I/! (5)[0] = 1^(^17! (2)[0]+il71(l)[0] i^(0)[0])+^(Hi(2)[0]4) (57) 2 16e~2 -5 1080 ' 2 /2 U2(5)[0] = — (-171(1)[0]--171(2)[0]): 2 189' Using the inverse DT (Definition 3), we get approximate solution for the IVPs (48)-(50) on the interval [0,fj: 8 Complexity 1 2 2 + e~2 3 4e~2 + 5 M, r (f) = 1 + f + -t + -f ' 72 1080 2 2 3 2 4 2 (58) u2Ji (f) = f - -f + —t + ----t 3 27 189 which is exactly the same approximate solution which has been obtained in [18]. The second step brings us to solving the given IVP on the interval [tl,t2\, where r2 is the minimal root of t - (l/2)e~f = tx, t2~ 0,620556. Now taking into account that both proportional delays qxt = (l/3)f and q2t = (l/2)f and also the time-dependent delay t - r1(t) = t - (l/2)e~f map the interval [tl,t2\ into the interval [0,tj, system (2) becomes "' (t- 2) m, = e u (59) 1 «< It ;7i(i)+2(f-iK7ig). Denote 2i{fc(0}[*i] = {H2 (fc) [fj}^ and 9){f(t)} [h] = {F2W[ti]}Zo> where f (t) = u'2(t - (1/2^). By application of Theorem 2 to corresponding terms, system (2) transformed at f0 = tx reads as (k +l)(k + 2) (k + 3) U1 (k + 3) [rj 2*etinsk-ico/x + k_l\ ,tv Zj 3 Z x )^(x + *-0[0]f ■H2 (fcJ^J+F^fc)^ (60) (k+ 1) (fc + 2) (fc+3)L72 (fc + 3) [fj - 00 / x + k + 3 (fc + 3)(fc + 2)(fc+l)^-• L/2 (x + fc + 3) [0] fe)X + 2 ^ [(f! - 1)5 (/) + 5 (/ - 1)] v 1' 1=0 /1 nfc-;oo /x + k_i\ ,tv •(-) I x )ulix + k-im(-i). (61) Now denote 3>{g(t)} [u, (t,)] = {G2 (k) [u, (tj]}^; then, according to Theorem 3, G2 (k) [ux (f:)] = /2/3\ , , I I (ux (fj)) for k>0 and Theorem 1 implies H (0)^] =G2(0)[u1(r1)] = ijUl (t,)2 H2 (k) [tl]=t G2 (I) [Ul (t,)} ■ Bki (U, (1) [fj,..., (62) 1=1 17J/C-Z+l)^]), forfc>l. Further denote e(t) = t - (l/2)e~f and 9){e{t)}[tA = {E2(k)[t1]}Zo- /(0 = («20<0(0 and, since e(fj) = 0, Theorem 1 in combination with Lemma 1 yields E2(k)[t1] =tl-S{k) + S{k-\) F2(0)[t1]=U2(l)[0]=0, 1 e-^j-lf Y k\ ' k>0, F2 W [fl] = I (' + 1)^2 tt + 1) [0] • Kl (E2 (1) [*l]> E2(k-l+ l)^]), forfc>l. (63) To get the initial data UjfcHrJ and L72(fc)[f1] for k = 0,1,2, we have to transform u{ (f) = 17; (0) [0] + I/,- (1) [0]f + L7; (2) [0]f2 + L7,.(3)[0]f3 + ..., at f1; i = 1,2. For A: = 0,1,2, we have (64) 17,-(/c)^] =17,.(0)[0]+17,.(1)[0]( k )t[ l-k 2 ^ 2-k . TT /nr^i/' 3 \+3-k + L7,.(2)[0]( k jtr+L7,.(3)[0]^ + . (65) L7i(0)[f1] = £[/,.(*)[()]£ k=0 Ui(l)[t1] = YJ(k+Wi(k+l)[0]tk, (66) U,-(2)[*1] = X k=0 k + 2 Ui(k + 2)[0]tk. The initial values at ti will be approximated by taking finite sums in computer evaluations of the infinite sums above. Observe that ux (fj) = Ux (0) [fj. Now let us compute the first few values of H2(k) [fj. Denote Bkl = Bkl (U1 (1) [fj,..., Ux (k - I + 1) [rj). Then, the first values of Bk; are Complexity 9 so,o = !> Su=L71(l)[f1].S0j0=L71(l)[f1], ß2,i = (D • Si,o + Ui (2) [*i] • §o,o = U, (2) ft], fc = 0: (3) [tj = V1 • u^) + Ml (tO). 1 / / 1 00 /t \% fc = 1: (4) [tj = - f e"2 f ^ ■ - ■ X (* + D^i (x + 1) [0] ^j S2,2 = t/1(l)[f1]-Su =([/!(!) [fj)2, (67) and the coefficients H2 for A: = = 0,1,2 are H2(0)[*i] = u1(t1), =|(«i (*i)r1/3- 2 . ßi.i = ^("l -1/3 ■u1(i)[t1], H2(2)[t1] =f(«l(*l))-1/3- 1 . S2,l " g("l (or 4/3 £ ' ß2,2 =f(«l(*l))"1/3- 9V 1 •(Uxd)^])2 +^-«i(^))+5-(«i(*i)r1/3-u1(i)[t1] + 2(1+*!)), ■2)[0] 1 / /1\2 00 / x + 2 \ /t \x 1 co . x ■(f) +^-3-I(x+lK/1(x+l)[0](f) + y-«i(j))+^(|(«i(ti)r1'3-(U1(2)[t1]) i(u1(t1)r4/3-(^(D[ti])2-ti-2(i+t1)2). (72) (68) Let us turn our attention to the first few values of F2(k)[t1]- Starting with E2(k) [rj, £,(0)^] =f1-Vt' =0, and for L77, we obtain k = 0: [/, £2(1)[*!] = l+V'1 = l+f1; , ir , 1 f, ^2(2)[M =--• — =--■ Now, Bw are flw (£2 (1) [fj,..., E2 (k - I + 1) [fj): so,o = !» Su =£2(l)-S0,o = 1 fj s2,i = (!)' si,o + £2 (2)' so,o = -y» B2j2 = £2(l)-Bu =(1+*1)2. Finally, coefficients F2 f°r = 0,1,2 are F2(0)[t1]=U2(l)[0]=0, F2(l)[t1] = 2[72(2)[0] -Su =2(1+0 F2 (2) = 2U2 (2) [0] • S2jl + 3U2 (3) [0] • S2>2 (69) (70) (71) !(3)Itll=Hä^+3r(x+3)[01(2)" +i(tI-i)-«I(T). fc = l: LT2(4)[t1]=i( ^ Jl/2(x + 4)[0](f) 1 / 1 CO /f \X fc = 2: C71 (5)[t1]=^J(X^5 )l/2 (x + 5)[0](|)" +s((t'-i)ii(x;2)^-«™(i)* l °° /(a + - £(x + !)[/! (x + l)[0](f-J -f1-2(l+f1)2 At this moment, we substitute H2 and F2 into systems (60) and (61). The next three coefficients at tx for U1 are (73) Using the inverse DT, again we get approximate solution for the IVPs (48), (49), and (50) on the interval [tut2]: 10 Complexity Table 1: Example 1, error analysis of u in [l,e]. Exact solution DT 5 DT 10 DT 25 Matlab DDENSD t el \u-el\ \u-el\ \u - el\ \u-el\ 1 2.7183 0 0 0 0 1.4296 4.1769 0.0000 6.4890E- 12 8.8818E-16 1.6976E-5 1.8591 6.4182 0.0017 1.0381E-8 8.8818E-16 3.6095E-5 2.2887 9.8622 0.0211 1.2409E-6 3.5527E-15 9.3438E-5 e 15.1543 0.1273 3.0578E-5 1.2861E-12 1.4012E-4 Table 2: Example 1, error analysis of u in [e,ee]. Exact solution DT 5 DT 10 DT 25 Matlab DDENSD t el \u-e'\ \u-e'\ \u-e'\ \u-e'\ e 15.1543 3.5527E- 15 3.5527E- 15 3.5527E- 15 0 5.8273 339.4331 32.3491 1.333 2.2737E- 13 0.0743 8.9363 7.6028E + 3 4.4755E + 3 397.5509 2.2638E-5 3.1741 12.0453 1.7029E + 5 1.5373E + 5 5.6777E + 4 0.9751 112.2467 ee 3.8143E + 6 3.7554E + 6 2.6576E + 6 2.0358E+3 3.1395E + 3 ulj2(t)=U1(0)[t1]+U1(l)[t1](t-t1) + U1(2) [t,] ■ (t-t1)2 + U1(3)[t1}(t-t1f + ..., u2j2{t)=U2{0)[t1]+U2{\)[t1\{t-t1) + U2{2) [tA ■ (t-t1f + U2(.3)[t1](t-t1)3 + .... (74) As the calculations are getting more complicated, all the calculations have been done numerically only. 3.4. Numerical Results and Discussion. Table 1 shows comparison of results for Example 1 obtained by DT algorithm with the orders of Taylor polynomials of the approximate solution N = 5,10,25 to results of Matlab function DDENSD in the interval [l,e]. Since the exact solution is known, absolute errors illustrate precision of each algorithm setting. All numbers are rounded to four decimal places. We see that DDENSD performs satisfactory well and DT for N= 10,25 does even better, whereas DT for N = 5 does not show satisfactory precision. Table 2 brings the same comparison in the second interval [e,ee]. We can observe a fast growth rate of the function values of the exact solution, which leads to the growth of absolute errors and loss of precision in all settings. It indicates that at the end of the considered interval [e, ee], the rate of precision would be better seen using relative errors. Implementation of DT in Matlab in case of Example 2 produces numerical results which are listed in Table 3. The results of DT with order of the Taylor polynomial N = 10 are compared to values obtained by DT combined with modified Adomian formula in [18] and to values produced by Matlab function DDENSD. First, we should say that the function DDENSD had difficulty at 0 where the value of the delayed argument t/2 was equal to the argument itself. Hence, to make DDENSD work, we replaced t/2 by t/2 - 10~16 in the second equation of (2). Our hypothesis is that the reason of the DDENSD failure is a combination of two facts: the second equation is neutral with respect to a proportional delay and the interval where the problem is considered contains 0. Second, we should mention that the numerical results for DDENSD were obtained by looking for approximate solutions on the whole interval [0,f2]- When trying to follow the method of steps, i.e., using DDENSD on [0, fj and then on [tlyt2], the results on the second interval [f1;f2] did not correspond to reality: there was a discontinuity in u2 at tx. Furthermore, we recall that the values taken from [18] have been computed using symbolic software Maple and the source code of the computation has been lost. Now, we can see a very good concordance of all algorithms in numerical values of the second component u2, while we observe a growing distance between the values of the first component ux computed by presented DT algorithm and values computed by the other two algorithms. As ux has exponential characteristics, we interpret the growing distance as growing lack of precision of DT algorithm which is based on approximation by Taylor polynomials. We suppose that dividing the intervals [0, t J and [t 1; t2] into smaller subintervals, i.e., refining the mesh grid, and applying the DT algorithm on those smaller intervals consecutively will improve the performance of the presented algorithm. Although it seems that the algorithm used in [18] shows better performance than the one presented in this paper, we cannot claim it with certainty as the source code got lost and we are not able to reproduce the data. Moreover, the approach used in [18] involves calculations of symbolic derivatives which makes it difficult to implement in numerical software like Matlab. Complexity 11 Table 3: Example 2, comparison of presented DT, DT in [18] and Matlab function DDENSD for ux and u2 in the first step for t e [0, t j ], and presented DT and DDENSD in the second step for t e [t1,t2]. Method DT 10 DT 10 Matlab DDENSD Matlab DDENSD DT [18] DT [18] t ul u2 ul u2 ul u2 0.00 1.0000 0.0000 1.0000 0.0000 1.0000 0.0000 0.05 1.0513 0.0024 1.0513 0.0024 1.0513 0.0024 0.10 1.1054 0.0093 1.1050 0.0093 1.1051 0.0093 0.15 1.1625 0.0203 1.1614 0.0203 1.1618 0.0203 0.20 1.2230 0.0348 1.2204 0.0348 1.2209 0.0348 0.25 1.2871 0.0524 1.2822 0.0524 1.2832 0.0524 0.30 1.3552 0.0726 1.3469 0.0726 1.3481 0.0726 0.35 1.4277 0.0951 1.4146 0.0951 1.4160 0.0951 0.3520 1.4306 0.0960 1.4174 0.0960 — — 0.3904 1.4896 0.1146 1.4717 0.1146 — — 0.4289 1.5514 0.1340 1.5280 0.1340 — — 0.4673 1.6161 0.1541 1.5864 0.1541 — — 0.5057 1.6839 0.1747 1.6469 0.1748 — — 0.5441 1.7548 0.1957 1.7098 0.1957 — — 0.5826 1.8290 0.2169 1.7750 0.2169 — — 0.6210 1.9065 0.2380 1.8428 0.2381 — — 4. Conclusion In the paper, we presented an algorithm which makes use of the differential transformation to initial value problems for systems of delayed or neutral differential equations with nonconstant delays. Two examples have been chosen to validate and test the algorithm. Numerical comparison of the presented semianalytical approach to Matlab function DDENSD brought interesting and promising results. Example 1 showed expected and reliable behaviour of the differential transform in the first step of the method of steps and expected deviation in the numerical results from values of the exact solution in the second step. Furthermore, we could observe a good concordance between the presented algorithm and DDENSD. After facing difficulties with DDENSD in Example 2, we could confirm a very good concordance of both differential transform and DDENSD in values of the component u2 which has a polynomial character on the considered intervals. On the other hand, we observed a growing discrepancy between the two methods in values of the component ux which has an exponential character. Our conclusion is that the disagreement is caused by large lengths of the intervals where the approximate solution is computed using the differential transform and that refining the mesh grid is necessary to obtain better performance. Further investigation will be focused on experimenting with different densities of mesh grids and studying convergence of the algorithm to find the optimal mesh grid. Numerical experiments will be focused on tuning the performance on problems with high complexity whose exact solutions are known and subsequently on applications to nonartificial real-life problems whose exact solutions are unknown. Data Availability The numerical data used to support the findings of this study are included within the article. Conflicts of Interest The authors declare that there are no conflicts of interest regarding the publication of this article. Acknowledgments The first author was supported by the project CEITEC 2020 (LQ1601) with financial support from the Ministry of Education, Youth and Sports of the Czech Republic under the National Sustainability Programme II, and by the grant FEKT-S-17-4225 of the Faculty of Electrical Engineering and Communication of Brno University of Technology. The Article Processing Charge was funded by Open Access Fund of Brno University of Technology. This support is gratefully acknowledged. Supplementary Materials We have included the commented Matlab source code for the first step of the method of steps in Example 1 as supplementary material. (Supplementary Materials) References [1] T. Kalmar-Nagy, G. Stepan, and F. C. Moon, "Subcritical Hopf bifucration in the delay equation model for machine tool vibrations," Nonlinear Dynamics, vol. 26, pp. 121-142, 2011. [2] D. Adak, N. Bairagi, and R. 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Applications of the Differential Transform to second-order half-linear Euler equations Zuzana Patfkova * Josef Rebenda t Abstract The purpose of the paper is to show applications of the differential transform to second-order half-linear Euler equations with and without delay. The case of proportional delay is considered. Finding a numerical solution to an initial value problem is reduced to solving recurrence relations. The outputs of the recurrence relations are coefficients of the Taylor series of the solution. Validity of the presented algorithm is demonstrated on concrete examples of initial value problems. Numerical results are compared with solutions produced by Matlab function "ddesd". AMS (MOS) subject classification: 34K28, 34K07, 34A45, 65L03, 65L05. Keywords: Half-linear Euler equation; differential transform; method of steps; differential equation with delay 1 Introduction Half-linear Euler type equations have been studied extensively in terms of the qualitative properties. However, combination of research methodologies is one of the successful generators of new ideas, results, and insights. The aim of this paper is to complement the theoretical results about qualitative behavior of solutions - asymptotic formulas or oscillatory properties - to half-linear Euler equations with and without delay with finding an approximate solution to the initial value problem numerically. Motivated by the current progress in research on the differential transform, the purpose of the paper is to investigate how the differential transform algorithm can be applied to half-linear Euler equations with a proportional delay and without delay. The half-linear equation without delay can be achieved as a transformation of partial differential equations that contain the so-called p-Laplacian Apu := div(||V-řt||p-2V-ří), 'Department of Mathematics, Faculty of Applied Informatics, Tomas Bata University in Zlín, Nad Stráněmi 4511, Zlín, 76005, Czech Republic (patikova@utb.cz). f Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická 2848/8, 61600 Brno, Czech Republic (rebenda@vutbr.cz). 1 where for u(x) = u{x\,...,xjv), N G N, the symbol Vw stands for the Hamilton nabla operator and div represents the divergence operator. Origins of p-Laplacian are described, for example, in the paper [2]. Accordingly, the history of p-Laplacian is closely linked to applications in the filtration of fluids through porous media and nonlinear non-Newtonian fluid dynamics. Another application can be found, for example, in the paper [1]. The p-Laplacian is used to model a non-homogenous diffusion to determine the height of a growing pile of noncohesive sand, where an ordinary differential equation arises in the limit case of "infinitely fast/slow" diffusion (see also [14]). The paper is organized as follows. In Section 2, we briefly summarize the results about qualitative properties of solutions to half-linear Euler equations with and without delay. Section 3 is devoted to theory and results on the differential transform. Sections 4 and 5 contain the main results of the paper. Numerical algorithms of the differential transform are adapted and applied to ordinary and delayed half-linear Euler equations. Concrete examples are illustrated by numerical results. Section 6 concludes the paper with a summary and outlines possibilities for further research. 2 Half-linear differential equations Half-linear differential equation of the second order is an equation of the form (r(i)$(x'))' + c(i)$(x) = 0, $(x) = |x|p_1sgnx, p > 1, (1) where r(t), c(t) are continuous functions and r(t) > 0. For p = 2, the equation (1) reduces to the second-order linear Sturm-Liouville differential equation (r(t)x')' + c(t)x = 0. Also from this point of view, the study of the properties of its generalized form (1) is a natural direction of research. The name "half-linear" has its origin in the fact that the space of solutions to (1) is homogeneous but not additive. The qualitative theory of half-linear differential equations has been studied extensively during the last decades. For the summary of the results up to 2005, we recommend the book [7]. More recent results can be found, for example, in [15, 32, 6, 26, 12, 16, 27] and references therein. From the qualitative point of view, half-linear differential equations of the form (1) can be divided into two classes. A half-linear equation is either oscillatory, which means that every its nontrivial solution has infinitely many zeros that form a sequence tending to infinity, or non-oscillatory, which means that every solution has constant sign in a neighbourhood of infinity. We recall that an oscillatory solution and a non-oscillatory solution to (1) cannot exist at the same time, which is a direct consequence of Sturm separation theorem [7, p. 16, Theorem 1.2.3]. 2 The subject of our interest in this paper is the second-order half-linear Euler equation (^'(t)rr+^a(t) = o, (2) as well as the second-order half-linear Euler equation with a proportional delay ((x'(t)n' + ^Tx«(At)=0, (3) where a > 0 is a quotient of two odd positive numbers, 7 G (0, 00) and A G (0,1). Euler equation (2) and its generalized forms belong to the most studied half-linear differential equations, see, for example [5, 8, 4, 10, 27]. For r(t) = 1, c(t) = an(l V = a + T the equation (1) is reduced to the Euler equation (2). Half-linear Euler equation is conditionally oscillatory with the oscillation constant / \ a+l that is, (2) is oscillatory if 7 > 7Q and non-oscillatory if 7 < 7Q. If 7 = 7Q, the equation (2) is also non-oscillatory and the two linearly independent solutions forming the solution space have the form xi(t) = t^TT, x2{t) ~ t°TT log^+i (£), where ~ means the asymptotic equivalence as t —> 00. It means that one of the solutions is known explicitly, whereas we only have an asymptotic formula for the second one. For details we refer to [7, Section 1.4.2]. Asymptotic formulas for the two linearly independent solutions are known also in the case 7 < 7Q (see [13]). For i = 1, 2 these are of the form x Xi{t) ~ txi" as t —>• 00, where \ are the zeros of the equation |A|1+- -A + 7 = 0. The Euler equation with a proportional delay (3) can be seen as a special case of the delayed half-linear equation (r(t)*(o:'(t)))' + c(t)*(a:(r(t))) = 0, (4) where r(t), c(t), r(t) are continuous functions on [to, 00), r(t) > 0, and t is a delay function satisfying r(t) < t, r'(t) > 0, lim r(t) = 00. In contrast to the non-delayed case (1), delayed half-linear equations may have oscillatory and non-oscillatory solutions simultaneously. A consequence 3 of this fact is that techniques applicable to half-linear equations without delay (1) often cannot be applied to delayed half-linear equations. In particular, only Riccati type inequality is available instead of the Riccati equation. Moreover, to prove that a solution is oscillatory is easier than to prove that it is non-oscillatory. If we choose r(i) = 1, c(t) = js+t, p—l = a, and r(t) = At in (4), we obtain the delayed Euler equation (3). Criteria providing conditions on 7 under which (3) has only oscillatory solutions were studied, for example, in [9] and [11]. 3 Differential Transform The Differential transform (DT) is a semi-analytical method based on Taylor's theorem. Its history dates back to 1970s to the work of G. E. Pukhov [17]. It has been shown that DT is convenient for solving a variety of initial value problems (IVPs), covering the range from ordinary to functional, partial and fractional differential equations [22, 29, 25, 28, 24, 19]. In particular, results on the differential equations with proportional, constant and non-constant delays can be found in [30], [23] and [21]. The differential transform of a real function u{i) at a point to G M that is analytic in a neighbourhood of to is V{u(t)}{to] = {U(k){t0}}%L0. (5) Here U(k)[to] is the kth component of the differential transform of the function u(t) at to, k G No, that is defined by (6) t=t0 The inverse differential transform of {U(k)[to]}^=0 is defined by 00 u(t) =V-1{{U(k)[t0]}^=0}[t0] = J2u(k)[t0](t-t0)k. (7) fe=0 In applications, the function u{i) is approximated by the finite sum N u(t) = J2u(k)[t0](t-t0)k. k=0 As we can see from (7), DT is related to the Taylor series. It means that the results about convergence of Taylor series may be used to decide on convergence of the DT algorithms. However, we refer to the particular paper [31] where the optimal general explicit a-priori error estimates are given. The following results will be used in the application sections 4 and 5. U(k)[t0 1 k~\ tdku(t) dtk 4 • Assume that u{i) is a real analytic function near to- Then (6) implies that ii"(t ii"'(t } U(0)[t0] = u(t0), U(l)[t0] = u'(t0), U(2)[t0] = -^L, U(3)[t0] = .... (8) The relationships (8) will be used by transforming initial conditions. • Assume that U(k)[to] is the kth component of the differential transform of the real analytic function u{i) at to. Then V{u'(t)}[t0] = {(k + l)U(k + l)[t0]}£=o- (9) Proof: Using (7), we can write j oo oo oo = ^£WM(*-*o)fe = J2ku(k)[t0](t-t0)k-1 = J2(k+l)u(k+i)[t0](t-t0)kn fc=0 k=l k=0 • [21] Assume that F(k)[to] is the kth component of the differential transform of the function /(t) at t0 and rel. If f(t) = tr, then F(k)[t0] = ( \y0-k (10) for all t such that |t-t0| < |t0|, where ' >- v ' - w Kk) k\ k\ and (r)fe represents the Pochhammer symbol. • [23] Assume that F(k), G(k) are the kth components of differential transforms of functions /(£), g(t) at a point to. Differential transform of a product f(i)g(i) at t0 is {fe J2F(l)G(k-l)\ . (11) '=0 J fe=0 • [18] Let g and / be real functions analytic near t0 and g(t0), respectively, and let h be the composition h(i) = (f o g)(t) = f(g(i)). De-note V{g{t)}[t0] = {G(k)}™=0, V{f(t)}[g(t0)} = {F(k)}™=0 and V{(f o g)(t)}[to] = {H(k)}^=0 the differential transforms of functions g, f and h at to, g(to) and to, respectively. Then the numbers H(k) in the sequence {H(k)}™=0 satisfy the relations H(0) = F(0) and fe H(k) = J2 F(l) ■ Bk,i (G(l),..., G(k - I + 1)) for k > 1, (12) 1=1 where Bk,i(xi,... ,Xk—i+i) are the partial ordinary Bell polynomials. 5 • [18] The partial ordinary Bell polynomials Bk,i(xi, ■ ■ ■, Xk—i+i), I = 1,2,..., k >l, satisfy the recurrence relation k-i+i ^ Bk,i(xi, ■ ■ ■, Xk-i+i) = XI ~£iBk-i,i-i(xi, ■ ■ ■ ,xk-i-i+2), i=i where Boo = 1 and Bk,o = 0 for k > 1. The first few polynomials Bk,i(xi, . ..) are: = 1 Bi,o = 0 = Xi B2,0 = 0 B2,l = x2 B2,2 B?,fl = 0 ■83,1 = x3 b32 = 2xxx2 S3,3 = (xi)3 (13) • [20] Let T>{f(t)}[t0] = {F(k)[t0]}f=0 be the differential transform of the function /(£) at to- Then the components F(k)[ti] of the differential transform T>{f(t)}[ti] = {F(k)[ti]}^_0 of f(i) at ti > to may be expressed as nWi] = Jt(k + J)(h-t0yF(k + j)[t0], k>o. (u) j=o ^ 3 ' 4 Application of the differential transform to the Euler equation Consider the initial value problem for the half-linear Euler equation of the form ((*T)'+^K*Q=0, x(t0) = a, x'(t0) = b, (15) in the case when a is a quotient of two positive odd numbers. The main goal of this section is to obtain image of the equation (15) under the differential transform at to. We apply the formulas introduced in the Section 6 3. Number of the applied formula appears in parentheses above the "=" sign. V{x(t)}® {X(k)}Zo, *(0)(=a, X(l)(i}6, V{x'(t)} {^ {(k + l)X(k + 1)}~ Q , V{(x'(t))a} (=} j£ Q (X(l)r-'Bfe,((2X(2),3X(3),...) j =: {Hi (*0}£=i Hi(o)= (p)(^(i)r, ^ {(fc + 1)^ + 1)}~ Q , {fe ~1 00 E (") (X(0)r-lBkAX(l),X(2),.. .)| =: {H2W}r=i > h2(o) = ^(x(o)r, v , 1 I (10) 1 ( ~Q " 1 „„, ,("v1)'»°"i""L=:{F,(",r-- Now, we use of the property (11) for transforming the product. The image of the equation (15) transformed at t = to reads as fe (k + l)Hi(k + l) + 'y^2F1(l)H2(k-l) = 0, keN0. (16) 1=0 We demonstrate the use of this result on a concrete example. Example 1 As a testing example, we take the Euler equation (2) with a = 3: We know that if 7 = 73 = (|)4, then xo(i) = f3/4 is a solution to the initial value problem ((x')3)' + -^fU3 = 0, x(l) = l, *'(!) = !■ (17) For brevity, we will use X(k) instead of A(/j)[l], k e No- Transformed equation (16) expands for (17) with a = 3,to = 1, 7 = (j) to 0=(k+l)J2 (,) (X(l))a-lBk+u(2X(2),...) ;=i ^ ' + 7E (?) E ( ■) W0))3-4-o(A(l),...) + 7(?) Q WO))3, 7 where the middle term applies only for k > 1. Taking into account (13), we can write the preceding equality in a more compact form fe+l /o\ 0 = (k + 1) J (X(1))^'S,+U(2X(2),...) ;=i ^ ' + 7E f",4) E f3)(^(0))3-JSfe_JjJ-(X(l),...), A- e N0. i=o v 1 ' j=0 u/ We start with k = 0: 1 • Q (X(1))2SM(2X(2),...) + 7(~Q4) Q (X(0))3 = 0 ^ 3(X(1))=2X(2) +7 = o => x(2) = = M. = - A. Then we substitute A; = 1: 2 (X(l))2S2jl(2X(2),...) + Q)x(l)B2j2(2X(2),...) + 7 ( (_04) (J) (X(0))2S1,1(X(1),...) + Q (X(0))3) = 0 2(3(X(1))2 • 3X(3) + 3X(1)(2X(2))2) + 7(3(X(0))2X(1) - 4(X(0))3) = 0 5 X^ = - 128- We continue with k = 2: 3 ( Q (X(l))2S3,i(2^(2), • • •) + Qx(l)S3j2(2X(2),...) + Q (X(1))°4,3(2X(2),...)) + 7(( Q4 f) (X(0))2S2jl(X(l),...) + (I )x(0)b2,2(x(1),...) + ( i4)(J)wo))2Bi,iWi)-.) + (24)(o)(*(o))3) =0, that is 3 (3(X(1))24X(4) + 3x(1)2 • 2X(2)3X(3) + (2X(2))3)) + 7 (j3(X(0))2X(2) + 3X(0)(X(1))2] - 4 • 3(X(0))2X(1) + t^l(X(0))3 45 Now recall that the exact solution to the initial value problem (17) is xo(t) = £3/4. Taylor series expansion of t3/4 at t0 = 1 is 1 + -(* - 1) - — (t -l)2 + —(t- l)3 - — (t - l)4 + 0((i - l)5). 4V ; 32v ; 128V ; 2048v u ; ; We can observe that using the differential transform algorithm we got the first several coefficients of Taylor expansion of the exact solution. The Taylor expansion is valid for \t — 1| < 1 since we used the formula (10) in the derivation of the transformed equation (16). Remark 1 A different approach to find recurrence relations for obtaining the coefficients of a Taylor series of the solution is presented in papers about the Parker-Sochacki method (see for example [3, 31]). The main idea is to transform an ordinary differential equation into a system of first order ordinary differential equations with nothing but polynomials on the righthand side. Such a method can be applied to a wide class of ordinary differential equations. However, describing the process of finding the polynomial form is not simple and the polynomial system might not be unique. A-priori error estimates presented in [31] can be applied especially to this polynomial form of the equation. Within this context, choosing the transformations y\ = x(t),y2 = x'(t),y^ = t~4,y4 = t,y§ = \/x'{t), the initial value problem (17) can be rewritten in the following polynomial form: y'i = V2, Z/i(l) = 1, y'2 = 1 /3\4 3 2 -3 (41 vi y-i 2/5. 2/2(1) = 3 ~ I' v'r -42/3 Vl 2/3(1) = 1, 1, 2/4(1) = 1, y'5 = 1 m4 3 4 3(4) yi y^ 2/5(1) = 4 ~ 3' 5 Application of the differential transform to the Euler equation with a proportional delay Consider the initial value problem {{x'{t)r)' + ^ri{x{\t)r = 0, x(t0) = a, x'(t0) = b, (18) with A G (0,1) and the initial function (t) = a + b(t-to), te(0,t0]=/o- (19) Let ti = |j and Jj = i e No. If t G Ii then At lies in We follow the process of combining the differential transform with the method of steps for delayed differential equations described in the paper [23]. For t G I\ = [to,t{\ we determine the solution of the initial value problem (18), (19) as xi(t). The differential transform of xi(i) at t0 will be Xi(k)[t0], 9 k e No- Since At for t e Ji falls into Jo, we substitute the initial function (19) for x(Xt) and rewrite equation (18) in the form ((x'(t))ay + ^LT(a + b(Xt-t0))a = 0, x(t0) = a, x'(t0) = b. (20) Because 4>{\t) =a + b(Xt - t0) = a + bt0(X - 1) + bX(t - t0), we have 2?l>(At)}[t0] = {a + bt0(X - 1), bX, 0,0,...}, and the kth component of the differential transform of ((Xt))a at t0 is fc V{( 0. 2=0 After we use the formula (11), the equation (20) transformed at to reads as 0 = (k + 1) E (fj (X1(l)[t0])Q-'Sfe+ljJ(2X1(2)[to], 3^(3) [t0] • • •) (21) + ?E ("V*)(to)—1-1 ( ° ) (a + i)T-k+l(bX)«-1. 1=0 ^ ' ^ ^ The initial conditions are transformed to X1(0)[t0] = a, X1(l)[t0] = b. Substitution for A; = 0,1,... into (21) provides recurrence relations from which one can successively calculate Xi(k)[t0] for k > 2. The solution on the interval Ii is then oo fe=0 In applications, we use a computing software to calculate the coefficients of the Taylor series. It means that this series as well as any other series will be truncated. The solution x\ will become an approximate solution. Notice that with a general initial function , equation (21) would have the form fe + l , s (Hl)^r \to\T-lBw{2XX(2)[t0],3*!(3)[t0]...) i=i ^ ' k ,1(t) = J2Xi(k)[t0](t-t0)h + 7E( ai l)(to)-a-1-lV{(cP(Xt)T}(k)lto] = 0. . I 1=0 10 Now we proceed with the second step. Take t G I2 = [ti,t2], denote X2(i) the approximate solution on I2, and let X2(k)[ti] for k > 0 be the differential transform of X2 at t\. Since At for t G I2 lies in ii, we substitute for x(Xt) the function xi(Xt) and rewrite equation (18) in the form ((x'(t))ay + -^I(x1(Xt))a = 0, x(t1)=x1(t1), x'(t1)=x'1(t1). (22) Since At = Ati + X(t — ti), we have V{Xt} = {Xh,X,0,0,...}. Recalling that Ati = to, we have the following expression for the kth component of the differential transform of xi(Xi) at t\: k Vix^XtMkfth] (=2) Ylxi(l)[^ti]Bk,i(X,0,0,...)=X1(k)[t0]Xk =: G(k)[h],k> 1, 1=1 X>{a;i(At)}(0)[ti] = Xi(0)[t0] = « =: G(0)[ti]. Next, for (xi(At))" we get v{(Xl(xt)rm[ti] (12)=(13) E (") (G(o)[t1]r-iBkAG(i)[t1},G(2)[t1], ...),*> o. 1=0 ^ ' Again, we use the product formula (11) and the equation (22) transformed at ti reads as 0 = (fc + 1) (Y) {X2{l)[t1\T-lBk+u(2X2(2)[h], 3X2(3)[h]...) (23) + T E f ~T ^ E f(G(0)[t1])--'Sfe_;j(G(l)[t1], G(2)[t!],...). According to (14), the initial conditions are 00 X2(0)[t1] = X^O)^} = E*i(*0[*o](*i -to)\ 00 X2(l)[ti] =X1(l)[t1] = J2(k+l)xi(k+l)ito}(ti-t0)k. k=0 The approximate solution x2 for t G I2 is then 00 a;2(t) = E^2(*;)[ti](t-ti)fe. fe=0 Further steps for t G Ii, i > 3 lead again to the recurrence relation (23). The only differences will appear in indeces of the Taylor coefficients (that is, X2 11 becomes Xi) and centres of the Taylor expansion (that is, t\ becomes U-i). The process of calculation of the Taylor coefficients in the ith step follows the pattern of the second step. Example 2 To demonstrate the described algorithm, we choose the following concrete values of the parameters: /3\4 3 a = 3, t0 = l, 7=(-J , a=l, b =-, A = 0.8. (24) Notice that the constants a, to and 7 have the exactly same values as in Example 1. Comparison with the solution xo(t) = £3/4 to the non-delayed problem (17) studied in Example 1 will allow us to observe the effect of the delay and the chosen initial function. This is important because it is not possible to find exact solution to the initial value problem (18), (19) with the constants (24) in terms of elementary functions. The equation (18) with the constants (24) becomes ((^'(i))3)'+(^)4^W0.8t))3=0, x(l) = l, z'(l) = ! (25) and the initial function is = l + te(0,l]. (26) Recalling the fact that tj = j*., the first step of the algorithm takes place on the interval [to,ii] = [1, whereas the second step takes place on the interval [^1,^2] = [^f, ^?]- The results of the simulation in Matlab, version 2019b, are shown in Table 1 and Figure 2. The first column of Table 1 presents the values of t in the interval [to, t2] where the comparison is done. In the second column we have values of the approximate solution to the IVP (25), (26) found by using the differential transform algorithm. Here x\ represents the solution on [to^i] and x2 the solution on t2\. In both cases, the order of the Taylor polynomial that represents the approximate solution is 5. We note that the accuracy influenced by the chosen order is good enough. The difference between the 5th order approximate solution and higher order approximate solutions on the observed interval is less than 10~6. The third column contains values of the approximate solution computed by the built-in Matlab function designed for solving delay differential equations "ddesd". The fourth column shows the difference between these two numerical solutions at given points. In the fifth column we present values of the exact solution to the initial value problem (17), that is, to the non-delayed half-linear Euler equation. At the end of the interval [toi^], we can observe that the solution to the half-linear Euler equation with delay tends to grow faster than the solution to the half-linear Euler equation without delay. This fact is illustrated also in Figure 2. Experimenting with other values of 7, we have noticed that neither "ddesd" nor a straightforward implementation of our algorithm can successfully calculate 12 Table 1: Comparison of DT (order 5) and Matlab x1(x2) ddesd •^ddesd %i x0 = t3/4 to = 1.0000 1.0000 1.0000 0.0 1.0000 1.0370 1.0621 1.0621 0.0147E - 3 1.0277 1.0741 1.1234 1.1235 0.1103E - 3 1.0551 h « 1.1111 1.1840 1.1843 0.3515E - 3 1.0822 1.1523 1.2591 1.2595 0.0004 1.1122 1.1934 1.3340 1.3339 -0.0002 1.1418 *2 « 1.2346 1.4089 1.4075 -0.0013 1.1712 the approximate solution on the interval [to, £2] if 7 increases to 1.9. Similar situation happens if we keep 7 at the constant value 1 and try to calculate on the interval [to, 3.0]: Matlab returns an error message. Such behavior suggests that more research in this direction is needed. 6 Conclusion We presented how the differential transform algorithm can be applied to obtain numerical solutions to second-order half-linear Euler equations without delay and with a proportional delay. The described algorithm includes a modification for different types of initial functions. Applicability of the algorithm is demonstrated on an example of the Euler equation without delay. The coefficients of the Taylor expansion obtained by the differential transform coincide with coefficients of Taylor expansion of the exact solution. Numerical simulation on an example of the delayed equation with concrete values of parameters was performed. The simulation values are in a good agreement with simulations produced by the Matlab routine "ddesd". All obtained results confirm that the presented algorithm is efficient and convenient for finding approximate solutions to the studied initial value problems. Our experiment shows that the differential transform method in combination with the method of steps is well applicable to delayed half-linear Euler equations. One of the advantages is that the obtained approximate solution is in the form of a Taylor polynomial. That is different from the outcome of the Matlab function "ddesd", where the result is a set of approximate function values of the solution. We also found out that both "ddesd" function and our procedure do not work on larger intervals in some cases, particularly with increasing 7. On the other hand, the combination of the differential transform and the method of steps can be theoretically elongated easily. The practical implementation of the algorithm, however, has to deal with the limits of division by numbers close to zero, especially when increasing the order of the Taylor polynomial. 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