Oscillation criteria for second order nonlinear neutral differential equations☆
Introduction
In this paper, we are concerned with the oscillatory behavior of second order nonlinear neutral delay differential equationswhere is a constant, , and .
By a solution of Eq. (1) we mean a continuous function , defined on , such that is continuously differentiable and satisfies (1) for In the sequel, we assume that solutions of Eq. (1) exist and can be continued indefinitely to the right. 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 Eq. (1) is said to be oscillatory if all its solutions oscillate.
Recently, an increasing interest in obtaining sufficient conditions for oscillatory or non-oscillatory behavior of different classes of differential and functional differential equations has been manifested. In particular, investigation of neutral differential equations is important since they are encountered in many applications in science and technology and are used, for instance, to describe distributed networks with lossless transmission lines, in the study of vibrating masses attached to an elastic bar, as well as in some variational problems, see [11]. It is well known [4] that the presence of a neutral term in a differential equation can cause oscillation, but it can also destroy oscillatory nature of a differential equation. In general, investigation of neutral differential equations is more difficult in comparison with ordinary differential equations, although certain similarities in the behavior of solutions of ordinary and neutral differential equations can be observed, see [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20].
In what follows, we briefly review several important oscillation results obtained for second order neutral differential equations. Grammatikopoulos et al. [9] established that conditionensures oscillation of a linear neutral differential equationSufficient conditions for the oscillation of solutions of a slightly more general neutral differential equationincluding the case when , were obtained by Džurina and Mihalı´ková [2]. By using the integral averaging method, Ruan [17] derived a number of general oscillation criteria for a nonlinear neutral differential equationwhereas Li [14] provided classification of non-oscillatory solutions of Eq. (2) and established necessary and/or sufficient conditions for the existence of eventually positive solutions. Ruan’s results for Eq. (2) have been further improved by Li and Liu [15] who exploited a generalized Riccati transformation. Interesting applications of the integral averaging technique to oscillation of several classes of nonlinear neutral differential equations can be found in the papers by Džurina and Lacková [1], Gai et al. [5], and Xu et al. [19]. In particular, the latter paper addresses the oscillation of a nonlinear neutral differential equationwhere. Recently, Shi and Wang [18] proved several oscillation criteria for Eq. (1), one of which we present below for the convenience of the reader. In what follows, we use the following notation: Theorem 1 [18, Theorem 2] Let the following conditions hold: for all , and is not identically zero for large t, , for all , and , , for , and has the sign of x and y if they have the same sign.
Suppose further that there exist functions , and satisfyingAssume also thatandIf there exists a function such thatandfor any , where , then Eq. (1) is oscillatory.
Very recently, Rogovchenko and Tuncay [16] established new oscillation criteria for a second order nonlinear differential equation with a damping termwithout an assumption that has been required in related results reported in the literature over the last two decades. The purpose of this paper is to strengthen oscillation results obtained for Eq. (1) by Shi and Wang [18] using a generalized Riccati transformation and developing ideas exploited by Rogovchenko and Tuncay [16]. In order to illustrate the relevance of our theorems, two examples are provided.
Section snippets
Main results
We say that a continuous function belongs to the class if:
- ()
and for ;
- ()
H has a continuous partial derivative with respect to the second variable satisfying, for some locally integrable continuous function h,
Theorem 2
Let conditions of Theorem 1 hold with replaced by, for , and , for .
Suppose that there exits a function such that, for some and for some ,
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Cited by (64)
Delay differential equation of fourth-order: Asymptotic analysis and oscillatory behavior
2022, Alexandria Engineering JournalCitation Excerpt :Besides theoretical interest and interesting problems, delay differential equations have many applications in applied science (see [1–3]). Furthermore, Works [9–13] were concerned with the development of oscillation criteria for second-order equations. While, [14–18] dealt with the study of oscillations for third/fourth-order equations.
New oscillation criteria for nonlinear delay differential equations of fourth-order
2020, Applied Mathematics and ComputationCitation Excerpt :Unusually, 4th-order equations appear in the theory of number, see [12]. Due to the mentioned interesting agents for the study of 4th-order differential equations, and due to the theoretical interest in generalizing and stretching some recognized results from those given for equations of first/second-order (see [3,8–11,13,18,19,23,31]), the research of oscillation of such equations has been given much attention. In the past few years, researchers have shown interest in examining the oscillation of the higher-order differential equations see [4,5,7,22,25,32,33], and there are several papers that are interested in studying the fourth-order differential equations, the reader can check the papers [1,6,15,16,20,24,27,30].
Oscillation criteria for even-order neutral differential equations
2016, Applied Mathematics LettersSome remarks on oscillation of second order neutral differential equations
2016, Applied Mathematics and Computation
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Research of the second author has been supported in part through the grant from the Faculty of Sciences and Technology of the University of Kalmar.