Nonlinear Analysis: Theory, Methods & Applications
Asymptotic behavior of nonoscillatory solutions to -th order nonlinear neutral differential equations
Introduction
In this paper, we are concerned with the asymptotic behavior of nonoscillatory solutions to the -th order nonlinear neutral differential equation of the form Recall that a nontrivial solution of Eq. (1) is called oscillatory if there exists a sequence of real numbers , diverging to , such that . Neutral differential equations have many important applications in science and technology and are used, for instance, to describe distributed networks with lossless transmission lines. It is known [6] that the presence of a neutral term in a differential equation can cause oscillation, and it also can destroy the oscillatory nature of the original differential equation. In general, the study of neutral differential equations brings more complications compared to that of ordinary differential equations, although in some cases certain similarities in the behavior of solutions of ordinary and neutral differential equations can be observed.
Recently, several authors discussed the existence of solutions to second order nonlinear differential equations and which behave like nontrivial linear functions at infinity; see, for instance, Constantin [1], [2], Lipovan [16], Mustafa and Rogovchenko [17], [18], [19], [20], Philos et al. [23], Rogovchenko and Rogovchenko [24], [25], Rogovchenko [26], Rogovchenko and Villari [27], Seifert [28], Serrin and Zou [29], Yin [31], Zhao [32].
Many interesting results regarding asymptotic properties of solutions of different classes of functional differential equations have been obtained by Dahiya and Singh [3], Dahiya and Zafer [4], Džurina [5], Graef and Spikes [8], Grammatikopoulos et al. [9], Kong et al. [12], Kulcsár [13], Lacková [14], Ladas [15], Naito [21], [22], Tanaka [30], and other authors.
In particular, Kong et al. [12] gave a classification of nonoscillatory solutions of an odd order linear neutral differential equation and established conditions for the existence of each type of nonoscillatory solution. Naito [21] proved that an -th order nonlinear neutral differential equation has a solution satisfying if and only if whereas Naito [22] derived a necessary and sufficient condition for a neutral differential equation to have a positive solution satisfying
Recently, Džurina [5] extended results of the second author [26] to a class of neutral differential equations establishing conditions under which all nonoscillatory solutions behave like linear functions as for some .
The purpose of this paper is to find sufficient conditions which guarantee that all nonoscillatory solutions of the general -th order nonlinear neutral differential equation (1) satisfy, for some real , In addition, for a particular case of Eq. (1) of the form we establish necessary and sufficient conditions for the existence of nonoscillatory solutions satisfying condition (3). Our theorems extend to Eq. (1) recent results on asymptotic behavior of nonoscillatory solutions of second order nonlinear neutral differential equations reported by Džurina [5] and complement research on asymptotic behavior of nonoscillatory solutions of functional differential equations reported by Dahiya and Singh [3], Dahiya and Zafer [4], Graef et al. [7] Graef and Spikes [8], Grammatikopoulos et al. [9], Kong et al. [12], Kulcsár [13], Ladas [15], Naito [21], [22], and Tanaka [30]. Examples are provided to illustrate the main results obtained in the paper.
Section snippets
Auxiliary lemma
Let . In what follows, we suppose that
(A1) , and there exist functions , such that either or where, for , the functions are positive, nondecreasing and (A2) , and ;
(A3) , and .
For , let
Continuation of solutions
The following result has independent interest and is used to assure that any nonoscillatory solution of Eq. (1) can be indefinitely continued to the right.
Theorem 3 Suppose that there exists a nonoscillatory solution of Eq.(1)defined on , which cannot be continued to the right beyond . (i) If satisfies(5), then . (ii) If satisfies(6), then .
Proof (i) Let be a nonoscillatory solution of Eq. (1) defined on , which cannot
Asymptotic behavior of solutions of Eq. (1)
Theorem 5 Suppose that(5)holds andThen any nonoscillatory solution of Eq.(1)satisfies(3), and there exist nonoscillatory solutions for which .
Proof Let be a nonoscillatory solution of Eq. (1) and be defined by (15). Then (20) holds, and Observing that one has An application of the Bihari inequality yields
Asymptotic behavior of solutions of Eq. (4)
In this section, we study asymptotic behavior of solutions of Eq. (4). In what follows, we suppose that
(B1) , and there exist functions , such that, for are nondecreasing, and either or (B2) , and ;
(B3) , and ;
(B4) if and have the same sign, then
Acknowledgements
The authors express their sincere gratitude to the referee for careful reading of the manuscript and valuable suggestions that helped to improve the paper. The research of the second author has been supported in part by the Abdus Salam International Centre for Theoretical Physics through Associate Membership and by the Faculty of Science and Technology of the University of Kalmar through a research grant.
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