Journal of Function Spaces
Volume 2016 (2016), Article ID 3954354, 5 pages
http://dx.doi.org/10.1155/2016/3954354

Research Article

On the Asymptotic Properties of Nonlinear Third-Order Neutral Delay Differential Equations with Distributed Deviating Arguments

Youliang Fu,¹ Yazhou Tian,¹ Cuimei Jiang,¹ and Tongxing Li2,3

¹Qingdao Technological University, Feixian, Shandong 273400, China
²LinDa Institute of Shandong Provincial Key Laboratory of Network Based Intelligent Computing, Linyi University, Linyi, Shandong 276005, China
³School of Informatics, Linyi University, Linyi, Shandong 276005, China

Received 23 April 2016; Accepted 23 June 2016

Academic Editor: Gestur Ólafsson

Copyright © 2016 Youliang Fu 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.

This paper is concerned with the asymptotic properties of solutions to a third-order nonlinear neutral delay differential equation with distributed deviating arguments. Several new theorems are obtained which ensure that every solution to this equation either is oscillatory or tends to zero. Two illustrative examples are included.

1. Introduction

In this paper, we consider the asymptotic properties of solutions to a class of differential equations of the form

where is a ratio of odd integers, , and . Throughout, the following hypotheses are tacitly supposed to hold:

),,,,,, , is not identically zero eventually, and for all;() and are nondecreasing functions forsatisfyingandfor,andfor, and, where .

During the last few decades, many researches have been done concerning the study of oscillation and asymptotic behavior of various classes of neutral differential equations, we refer the reader to the monograph [1], the papers [2–11], and the references cited there. The investigation of asymptotic behavior of (1) is important for practical reasons and the development of asymptotic theory; see Wang [10]. Tian et al. [9] explored asymptotic properties of (1) assuming that conditions, , and hold. Very recently, applying the Riccati transformation and Lebesgue’s monotone convergence theorem, Candan [5] established several oscillation criteria for a class of second-order neutral delay differential equations with distributed deviating arguments. Motivated by the method reported in the paper by Candan [5], the aim of this paper is to derive some new results on the oscillation and asymptotic behavior of solutions to (1) which can be applied in the case whereas well. These criteria provide answers to a question posed in [9, Remark 4.4].

We use the notation. By a solution to (1) we mean a nontrivial function satisfying (1) which possesses the properties and for . The focus of this paper is restricted to those solutions of (1) which have the property for all. A solutionof (1) is termed oscillatory if it does not have the largest zero on the interval;is said to be nonoscillatory if it is either eventually positive or eventually negative. In what follows, all functional inequalities are assumed to hold for all sufficiently large.

2. Auxiliary Lemmas

In order to establish our main results, we need the following auxiliary lemmas which are extracted from the paper by Agarwal et al. [2] and the paper by Tian et al. [9].

Lemma 1 (see [9, Lemma 2.1]). Let hypotheses () and () be satisfied and suppose that is an eventually positive solution of (1). Then, for all sufficiently large , satisfies either

(I), , , and

or

(II) , , , and .

Lemma 2 (see [2, Lemma ]). If satisfies case (I), then eventually.

Lemma 3 (see [9, Lemma ]). Suppose that is an eventually positive solution of (1). If satisfies case (II), then provided that

3. Main Results

For a compact presentation of our results, we use the following notation:

where is well defined.

Theorem 4. Let hypotheses (), (), and (2) be satisfied. Then all solutions of (1) either are oscillatory or converge to zero asymptotically if

Proof. Suppose to the contrary that is a nonoscillatory solution to (1). Without loss of generality, it may be assumed that is an eventually positive solution of (1). Then there is a such that, for , , , and , for and , respectively. In view of (1), we have

On the basis of Lemma 1, we observe that satisfies either case (I) or case (II) for.
Let satisfy case (I). It follows from the definition of that

which yields

and so

Substitution of (8) into (5) and the definition of imply that

Integrating (9) from to , we arrive at

Taking into account that and , there exists a constant such that . Therefore, we deduce that

and hence

which is a contradiction with (4).
Let satisfy case (II). Then when using Lemma 3. The proof is complete.

Now, we establish some oscillation criteria for (1) by utilizing the Riccati transformation and Lebesgue’s monotone convergence theorem. To this end, we give the following lemmas.

Lemma 5. Let be an eventually positive solution of (1) and let satisfy case (I). Define the Riccati transformation by

Then

Proof. By , we conclude that

which yields

It follows from (13) and case (I) that . Differentiation of (13) and applications of (9), (13), (16), and Lemma 2 imply that

which completes the proof.

Define a sequence of functions by and

where are well defined. By induction, for and.

Lemma 6. Let be an eventually positive solution of (1) and suppose that satisfies case (I). Then for , and there exists a function such that, for , and

where and are as in (13) and (18), respectively.

Proof. Integrating (14) from to , we deduce that

For every fixed , we claim that

If (21) does not hold, then, for every fixed ,

which is a contradiction to . It follows now from (14) that there exists a constant such that. Taking into account (21) and the condition , . An application of (20) yields

By induction, for and . Hence, when using the fact that the sequence is nondecreasing and bounded above. Passing to the limit as in (18) and applying Lebesgue’s monotone convergence theorem, one arrives at (19). The proof is complete.

Theorem 7. Let hypotheses (), (), and (2) be satisfied. If

then every solution of (1) either is oscillatory or satisfies .

Proof. Without loss of generality, suppose that is an eventually positive solution of (1). By virtue of Lemma 1, satisfies either case (I) or case (II) eventually. Assume first that satisfies case (I). Proceeding as in the proof of Lemma 6, we obtain (23) and so

Let . Then . Combining (24) and (25), we deduce that

However, an application of the inequality (see [3])

yields

and so a contradiction is presented.
Assume now that satisfies case (II). It follows from Lemma 3 that . This completes the proof.

Theorem 8. Let hypotheses (), (), and (2) be satisfied and suppose that is as in (18). If for some ,

then all solutions of (1) either are oscillatory or tend to zero asymptotically.

Proof. Assume the opposite. Let be an eventually positive solution of (1). By virtue of Lemma 1, satisfies either case (I) or case (II) eventually. Suppose first that satisfies case (I). Define by (13). Using Lemma 2 and the monotonicity of , we conclude that, for ,

which yields

Hence, we deduce that

On the other hand, by Lemma 6, for , and so

which contradicts (29).
Assume now that satisfies case (II). By virtue of Lemma 3, . The proof is complete.

Remark 9. Our results complement and improve those obtained by Tian et al. [9] since these results can be applied to (1) in the case where .

4. Examples

The following examples are included to show applications of the results obtained in this work.

Example 1. For and , consider the nonlinear differential equation

where is the quotient of odd integers. Let , , , , , , , , , and . It follows from Theorem 4 that every solutionof (34) either is oscillatory or converges to zero asymptotically. Observe that results obtained in [9] cannot be applied to (34) in the case when .

Example 2. For, consider the differential equation

Let , , , , , , , , , , and . Then

It is not hard to verify that hypotheses , , and (2) hold; , and

Hence

An application of Theorem 7 implies that all solutions of (35) either are oscillatory or satisfy.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This research was supported in part by NNSF of China (Grants nos. 61503171, 61403061, and 11447005), CPSF (Grant no. 2015M582091), DSRF of Linyi University (Grant no. LYDX2015BS001), and the AMEP of Linyi University, China.

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