Abstract and Applied Analysis

Volume 2014 (2014), Article ID 143614, 8 pages

http://dx.doi.org/10.1155/2014/143614

Research Article

Oscillatory Behavior of Second-Order Nonlinear Neutral Differential Equations

Tongxing Li¹ and Yuriy V. Rogovchenko²

¹Department of Mathematics, Linyi University, Linyi, Shandong 276005, China

²Department of Mathematical Sciences, University of Agder, P.O. Box 422, 4604 Kristiansand, Norway

Received 28 March 2014; Accepted 16 April 2014; Published 11 June 2014

Academic Editor: Qi-Ru Wang

Copyright © 2014 Tongxing Li and Yuriy V. Rogovchenko. 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.

Abstract.

We study oscillatory behavior of solutions to a class of second-order nonlinear neutral differential equations under the assumptions that allow applications to differential equations with delayed and advanced arguments. New theorems do not need several restrictive assumptions required in related results reported in the literature. Several examples are provided to show that the results obtained are sharp even for second-order ordinary differential equations and improve related contributions to the subject.

1. Introduction

This paper is concerned with the oscillation of a class of second-order nonlinear neutral functional differential equations

where . The increasing interest in problems of the existence of oscillatory solutions to second-order neutral differential equations is motivated by their applications in the engineering and natural sciences. We refer the reader to [1–21] and the references cited therein.

We assume that the following hypotheses are satisfied:

is a quotient of odd natural numbers, the functions , and ;the functions and

the function satisfies

for all and there exists a positive continuous function defined on such that

By a solution of (1), we mean a function defined on for some such that and are continuously differentiable and satisfies (1) for all . In what follows, we assume that solutions of (1) exist and can be continued indefinitely to the right. Recall that a nontrivial solution of (1) is said to be oscillatory if it is not of the same sign eventually; otherwise, it is called nonoscillatory. Equation (1) is termed oscillatory if all its nontrivial solutions are oscillatory.

Recently, Baculíková and Džurina [6] studied oscillation of a second-order neutral functional differential equation

assuming that the following conditions hold:

, , , and ; and ;, , and .

They established oscillation criteria for (5) through the comparison with associated first-order delay differential inequalities in the case where

Assuming that

Han et al. [9], Li et al. [15], and Sun et al. [20] obtained oscillation results for (5), one of which we present below for the convenience of the reader. We use the notation

Theorem 1 (cf. [9, Theorem 3.1] and [20, Theorem 2.2]). Assume that conditions and (7) hold. Suppose also that and for all . If there exists a function such that

then (5) is oscillatory.

Replacing (6) with the condition

Baculíková and Džurina [7] extended results of [6] to a nonlinear neutral differential equation

where and are quotients of odd natural numbers. Hasanbulli and Rogovchenko [10] studied a more general second-order nonlinear neutral delay differential equation

assuming that , , , and (6) holds. To introduce oscillation results obtained for (1) by Erbe et al. [8], we need the following notation:

We say that a continuous function belongs to the class if

(i) for and for ;(ii) has a nonpositive continuous partial derivative with respect to the second variable satisfying

for some and for some .

Theorem 2 (see [8, Theorem 2.2, when ]). Let conditions (10) and hold. Suppose that , , and for all . If there exists a function such that, for all sufficiently large ,

then (1) is oscillatory.

Theorem 3 (see [8, Theorem 2.2, case ]). Let conditions (10) and be satisfied. Suppose also that , , and for all . If there exists a function such that, for all sufficiently large and for some ,

then (1) is oscillatory.

Assuming that

Li et al. [16] extended results of [10] to a nonlinear neutral delay differential equation

where is a ratio of odd natural numbers. Han et al. [9, Theorems 2.1 and 2.2] established sufficient conditions for the oscillation of (1) provided that (17) is satisfied, , and

Xu and Meng [21] studied (1) under the assumptions that (17) holds, , and

obtaining sufficient conditions for all solutions of (1) either to be oscillatory or to satisfy ; see [21, Theorem 2.3]. Saker [17] investigated oscillatory nature of (1) assuming that (17) is satisfied,

and

for some , where . Li et al. [12] studied oscillation of (1) under the conditions that (17) holds, and are strictly increasing, , and

Li et al. [13] investigated (1) in the case where hold, , , and . In particular, sufficient conditions for all solutions of (1) either to be oscillatory or to satisfy were obtained under the assumptions that (17) holds and ; see [13, Theorem 3.8]. Sun et al. [19] established several oscillation results for (1) assuming that , , (17), and (23) are satisfied. The following notation is used in the next theorem:

Theorem 4 (see [19, Theorem 4.1]). Let conditions , , and (17) be satisfied. Assume also that , , and for all . Suppose further that there exist functions and such that , ,

Then (1) is oscillatory.

Our principal goal in this paper is to analyze the oscillatory behavior of solutions to (1) in the case where (17) holds and without assumptions , (19)–(23), and .

2. Oscillation Criteria

In what follows, all functional inequalities are tacitly assumed to hold for all large enough, unless mentioned otherwise. We use the notation

A continuous function is said to belong to the class if

(i) for and for ;(ii) has a nonpositive continuous partial derivative with respect to the second variable satisfying

for some .

Theorem 5. Let all assumptions of Theorem 2 be satisfied with condition (10) replaced by (17). Suppose that there exists a function such that

If there exists a function such that, for all sufficiently large ,

then (1) is oscillatory.

Proof. Let be a nonoscillatory solution of (1). Without loss of generality, we may assume that there exists a such that , , and for all . Then for all , and by virtue of

the function is strictly decreasing for all . Hence, does not change sign eventually; that is, there exists a such that either or for all . We consider each of the two cases separately.

Case  1. Assume first that for all . Proceeding as in the proof of [8, Theorem 2.2, case ], we obtain a contradiction to (15).

Case  2. Assume now that for all . For , define a new function by

Then for all , and it follows from (30) that

for all . Integrating (32) from to , , we have

Passing to the limit as , we conclude that

which implies that

Thus,

Consequently, there exists a such that, for all ,

Differentiating (31) and using (30), we have, for all ,

Hence, by (31) and (38), we conclude that

for all . Multiplying (39) by and integrating the resulting inequality from to , we obtain

In order to use the inequality

see Li et al. [16, Lemma 1 (ii)] for details; we let

Then, by virtue of (40), we conclude that

which contradicts (29). This completes the proof.

Theorem 6. Let all assumptions of Theorem 3 be satisfied with condition (10) replaced by (17). Suppose further that there exists a function such that (28) holds. If there exists a function such that, for all sufficiently large ,

then (1) is oscillatory.

Proof. The proof is similar to that of Theorem 5 and hence is omitted.

Theorem 7. Let conditions (10) and be satisfied, , , and . Assume that there exists a function such that, for all sufficiently large ,

If there exists a function such that, for all sufficiently large ,

then (1) is oscillatory.

Proof. Without loss of generality, assume again that (1) possesses a nonoscillatory solution such that , , and on for some . Then, for all , (30) is satisfied and . It follows from (10) that there exists a such that for all . By virtue of (30), we have

Since

we conclude that

For , define a new function by

Then for all , and the rest of the proof is similar to that of [8, Theorem 2.2, case ]. This completes the proof.

Theorem 8. Let conditions (10) and be satisfied. Suppose also that , , , and there exists a function such that (45) holds for all sufficiently large . If there exists a function such that, for some ,

then (1) is oscillatory.

Proof. The proof runs as in Theorem 7 and [8, Theorem 2.2, case ] and thus is omitted.

Theorem 9. Let all assumptions of Theorem 7 be satisfied with condition (10) replaced by (17). Suppose that there exist a function and a function such that

and, for all sufficiently large ,

Then (1) is oscillatory.

Proof. Without loss of generality, assume as above that (1) possesses a nonoscillatory solution such that , , and on for some . Then, for all , (30) is satisfied and . Therefore, the function is strictly decreasing for all , and so there exists a such that either or for all . Assume first that for all . As in the proof of Theorem 7, we obtain a contradiction with (46). Assume now that for all . For , define by (31). By virtue of ,

The rest of the proof is similar to that of Theorem 5 and hence is omitted.

Theorem 10. Let all assumptions of Theorem 8 be satisfied with condition (10) replaced by (17). Suppose that there exists a function such that, for all sufficiently large ,

Then (1) is oscillatory.

Proof. The proof resembles those of Theorems 5 and 9.

Remark 11. One can obtain from Theorems 5 and 6 various oscillation criteria by letting, for instance,

Likewise, several oscillation criteria are obtained from Theorems 7–10 with

3. Examples and Discussion

The following examples illustrate applications of theoretical results presented in this paper.

Example 1. For , consider a neutral differential equation

where and are constants. Here, , , , , , and . Let and . Then and, for all sufficiently large and for all satisfying , we have

On the other hand, letting and , we observe that condition (15) is satisfied for . Hence, by Theorem 5, we conclude that (58) is oscillatory provided that . Observe that results reported in [9, 12, 17, 21] cannot be applied to (58) since and conditions (19)–(22) fail to hold for this equation.

Example 2. For , consider a neutral differential equation

where is a constant. Here, , , , , , and . Let and . Then . Hence,

whenever . Let and . Then (16) is satisfied for . Therefore, using Theorem 6, we deduce that (60) is oscillatory if , whereas Theorems 1 and 4 yield oscillation of (60) for , so our oscillation result is sharper.

Example 3. For , consider the Euler differential equation

where is a constant. Here, , , , , and . Choose and . Then , and so

provided that . Let and . Then (15) holds for . It follows from Theorem 5 that (62) is oscillatory for , and it is well known that this condition is the best possible for the given equation. However, results of Saker [17] do not allow us to arrive at this conclusion due to condition (22).

Remark 12. In this paper, using an integral averaging technique, we derive several oscillation criteria for the second-order neutral equation (1) in both cases (10) and (17). Contrary to [9, 12, 15, 17, 19–21], we do not impose restrictive conditions and (19)–(23) in our oscillation results. This leads to a certain improvement compared to the results in the cited papers. However, to obtain new results in the case where (17) holds, we have to impose an additional assumption on the function ; that is, . The question regarding the study of oscillatory properties of (1) with other methods that do not require this assumption remains open at the moment.

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contribution

Both authors contributed equally to this work and are listed in alphabetical order. They both read and approved the final version of the paper.

Acknowledgments

The authors thank the referees for pointing out several inaccuracies in the paper. The research of the first author was supported by the AMEP of Linyi University, China.

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