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Abstract and Applied Analysis

Volume 2014, Article ID 618183, 4 pages

http://dx.doi.org/10.1155/2014/618183
Research Article

Asymptotic Properties of Solutions to Third-Order Nonlinear Neutral Differential Equations

1Qingdao Technological University, Feixian, Shandong 273400, China

2Department of Mathematics, Qufu Normal University, Qufu, Shandong 273165, China

3College of Mathematics and Information Science, Xinyang Normal University, Xinyang 464000, China

Received 26 March 2014; Accepted 12 May 2014; Published 25 May 2014

Academic Editor: Tongxing Li

Copyright © 2014 Qi Li et al. 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

The aim of this work is to discuss asymptotic properties of a class of third-order nonlinear neutral functional differential equations. The results obtained extend and improve some related known results. Two examples are given to illustrate the main results.

1. Introduction

In this work, we study the asymptotic behavior of solutions of the third-order neutral differential equation We always assume that the following conditions hold:(H1) ;(H2) , , , ;(H3) , , for all ;(H4) , ;(H5) .

Set . By a solution of (1), we mean a nontrivial function , , which has the properties , , and and satisfies (1) on . We consider only those solutions of (1) which satisfies for all . We assume that (1) possesses such a solution. A solution of (1) is called oscillatory if it has arbitrarily large zeros on ; otherwise, it is called nonoscillatory.

Recently, great attention has been devoted to the oscillation of various classes of differential equations. See, for example, [119]. Hartman and Wintner [1] and Erbe et al. [3] studied the third-order differential equation

Paper [5] studied the oscillation of third-order trinomial delay differential equation

Li et al. [7] discussed (1) with and . Han [8] examined the oscillation of (1) with .

In this work, we establish some oscillation criteria for (1) which extend and improve the results in [7, 8].

2. Main Results

In the following, all functional inequalities considered are assumed to hold eventually for all large enough. Without loss of generality, we deal only with the positive solutions of (1).

Theorem 1. Suppose that If for some function , for all sufficiently large , one has where then all solutions of (1) are oscillatory or convergent to zero asymptotically.

Proof. Assume that is a positive solution of (1). Based on condition , there are two possible cases:(1) , , , ;(2) , , , .

First, consider that satisfies (1). We have From (1), , and , we get Define a function by we obtain . Then By the proof of [7, Theorem 2.1], we have where is defined as in (6). We obtain That is, which contradicts (5). Assume that case (2) holds. Using the similar proof of [8, Lemma 4], we can get . This completes the proof.

Based on Theorem 1, we present a Kamenev-type criterion for (1).

Theorem 2. Assume that (4) holds. If for some function , for all sufficiently large , one has then all solutions of (1) are oscillatory or convergent to zero asymptotically.

Proof. Assume that is a positive solution of (1). Then by the proof of Theorem 1, we have cases (1) and (2). Let case (1) hold. Proceeding as in the proof of Theorem 1, we have (11). Then we have That is, which contradicts (14). Assume that case (2) holds. We can get . The proof is completed.

Next, we present a Philos-type criterion for (1). Let We say that a function belongs to a function class , if it satisfies(i) , ; , ;(ii) has a continuous and nonpositive partial derivative on with respect to the second variable, and such that

Theorem 3. Assume that (4) holds. If for some function , for all sufficiently large , one has where is defined as in (6), , and then all solutions of (1) are oscillatory or convergent to zero asymptotically.

Proof. Assume that is a positive solution of (1), and has the case of (1); is defined as in (9). Then Let , we have We obtain That is, which contradicts (19). Assume that (2) holds. We can get . The proof is completed.

3. Examples

In this section, we will present two examples to illustrate the main results.

Example 1. Consider the third-order nonlinear neutral differential equation: where , and . Let . It follows from Theorem 1 that every solution of (25) is oscillatory or convergent to zero asymptotically.

Example 2. Consider the third-order nonlinear neutral differential equation: where , . We have we see that (4) and (H1)–(H5) hold. Let , . Then . It follows, from Theorem 3, that the solutions of (26) are oscillatory or convergent to zero asymptotically.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This research was partially supported by the NNSF of China (11171178).

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