ISRN Mathematical Analysis

VolumeΒ 2011Β (2011), Article IDΒ 485203, 9 pages

http://dx.doi.org/10.5402/2011/485203

## Existence for Nonoscillatory Solutions of Higher-Order Nonlinear Differential Equations

^{1}Department of Basic Courses, Qingdao Technological University (Linyi), Feixian 273400, Shandong, China^{2}Department of Mathematics, Qufu Normal University, Qufu 273165, Shandong, China

Received 8 August 2011; Accepted 22 September 2011

Academic Editor: Z.Β Dosla

Copyright Β© 2011 Yazhou Tian and Fanwei Meng. 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 existence of nonoscillatory solutions of the higher-order nonlinear differential equation , where are integers, is studied. Some new sufficient conditions for the existence of a nonoscillatory solution of above equation are obtained for general which means that we allow oscillatory . In particular, our results improve essentially and extend some known results in the recent references.

#### 1. Introduction

Consider the higher-order nonlinear neutral differential equation With respect to (1.1), throughout, we shall assume the following:(i) are integers, ,(ii).

Let . By a solution of (1.1), we mean a function for some which has the property that and and satisfies (1.1) on .

A nontrivial solution of (1.1) is called oscillatory if it has arbitrarily large zeros, and, otherwise, it is nonoscillatory.

The existence of nonoscillatory solutions of higher-order nonlinear neutral differential equations received much less attention, which is due mainly to the technical difficulties arising in its analysis.

In 1998, Kulenovic and Hadziomerspahic [1] investigated the existence of nonoscillatory solutions of second-order nonlinear neutral differential equation where is a constant.

In 2006, Zhang and Wang [2] investigated the second neutral delay differential equation with positive and negative coefficients: where . By using Banach contraction mapping principle, they proved the following theorem which extends the results in [1].

Theorem A. ([2, Theoremββ2.3]). *Assume that** and satisfy local Lipschitz condition and for; ** is eventually nonnegative for every ;** hold **if one of the following two conditions is satisfied:** eventually, and ,** eventually, and ,**where , , then (1.1) has a nonoscillatory solution.*

In 2007, Zhou [3] studies the existence of nonoscillatory solution of the following second-order nonlinear differential equation. where . By using Krasnoselskiiβs fixed point theorem, they proved the following theorem.

Theorem B. ([3, Theoremββ1]). *Assume that there exist nonnegative constants and such that , . Further, assume that
**
Then (1.1) has a bounded nonoscillatory solution.*

In this paper, by using Krasnoselskiiβs fixed point theorem and some new techniques, we obtain some sufficient conditions for the existence of a nonoscillatory solution of (1.1) for general which means that we allow oscillatory . Meanwhile, we extend the main results of [2, 3].

#### 2. Main Result

The following fixed point theorem will be used to prove the main results in this section.

Lemma 2.1 (see [3, Krasnoselskiiβs fixed point theorem]). *Let be a Banach space, let be a bounded closed convex subset of , and let , be maps of into such that for every pair . If is a contraction and is completely continuous, then the equation
**
has a solution in .*

Theorem 2.2. *Assume that there exist nonnegative constants and such that , . Further, assume that
**
Then (1.1) has a bounded nonoscillatory solution. *

*Proof. *By interchanging the order of integral, we note that (2.2) is equivalent to
By (2.3), we choose sufficiently large such that
where .

Let be the set of all continuous functions with the norm . Then is a Banach space. We define a bounded, closed, and convex subset of as follows:

Define two maps and as follows:

(i) We shall show that for any , .

In fact, , and , we get

Furthermore, we have
Hence,
Thus, we have proved that for any .

(ii) We shall show that is a contraction mapping on .

In fact, for and , we have
where . This implies that
Since , we conclude that is a contraction mapping on .

(iii) We now show that is completely continuous.

First, we will show that is continuous. Let be such that as . Because is closed, . For , we have
Since as for , by applying the Lebesgue dominated convergence theorem, we conclude that . This means that is continuous.

Next, we show that is relatively compact. It suffices to show that the family of functions is uniformly bounded and equicontinuous on . The uniform boundedness is obvious. For the equicontinuity, according to Levitan's result [4], we only need to show that, for any given , can be decomposed into finite subintervals in such a way that on each subinterval all functions of the family have change of amplitude less than . By (2.3), for any , take large enough so that
Then, for , ,
For , ,
where .

Then there exists such that
For any , , it is easy to see that
Therefore, is uniformly bounded and equicontinuous on , and hence is relatively compact. By Lemma 2.1, there is such that . It is easy to see that is a nonoscillatory solution of (1.1). The proof is complete.

Theorem 2.3. *Assume that and (2.2) holds. Then (1.1) has a bounded nonoscillatory solution.*

*Proof. *We choose positive constants such that . . Choosing sufficiently large such that
where .

Let be the set as in the proof of Theorem 2.2. We define a bounded, closed, and convex subset of as follows:

Define two maps and as follows:

(i) We shall show that for any , .

In fact, for every , and , we get

Thus, we have proved that . Since , we get that is a contraction mapping. We also can prove that is uniformly bounded and equicontinuous on , and hence is relatively compact. So by Lemma 2.1, there is such that . That is,
It is easy to see that is a bounded nonoscillatory solution of (1.1).

The proof is complete.

Theorem 2.4. *Assume that and (2.2) holds. Then (1.1) has a bounded nonoscillatory solution.*

*Proof. *We choose positive constants such that . . Choosing sufficiently large such that
where .

Let be the set as in the proof of Theorem 2.2. We define a bounded, closed, and convex subset of as follows:

Define two maps and as follows:

(i) We shall show that for any , .

In fact, for every and , we get

Thus, we have proved that . Since , we get is a contraction mapping. We also can prove that is uniformly bounded and equicontinuous on , and, hence, is relatively compact. So by Lemma 2.1, there is such that . That is,
It is easy to see that is a bounded nonoscillatory solution of (1.1).

The proof is complete.

*Remark 2.5. *If we let in Theorem 2.2, we get the Theorem 1 in [3]. In the case where , , Theorem 2.2 improves essentially Theorem 2.2 in [5].

*Remark 2.6. *The conditions of Theorem 2.4 relaxing the hypotheses () of Theorem 3 in [2].

*Remark 2.7. *Theorems 2.3 and 2.4 improve essentially Theorem 3 in [2], we allow that are oscillatory.

#### Acknowledgments

This research was supported by Natural Science Foundations of Shandong Province of China (ZR2009AM011 and ZR2009AQ010) and Doctor of Ministry of Education (20103705110003).

#### References

- M. R. S. Kulenovic and S. Hadziomerspahic, βExistence of nonoscillatory solution of second order linear neutral delay equation,β
*Journal of Mathematical Analysis and Applications*, vol. 228, no. 2, pp. 436β448, 1998. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Z. Y. Zhang and X. X. Wang, βThe oscillatory and nonoscillatory criteria of second nonlinear neutral equation,β
*Journal of Systems Science and Mathematical Sciences*, vol. 26, no. 3, pp. 325β334, 2006 (Chinese). View at Google Scholar Β· View at Zentralblatt MATH - Y. Zhou, βExistence for nonoscillatory solutions of second-order nonlinear differential equations,β
*Journal of Mathematical Analysis and Applications*, vol. 331, no. 1, pp. 91β96, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - B. M. Levitan, βSome questions of the theory of almost periodic functions I,β
*Uspekhi Matematicheskikh Nauk*, vol. 2, no. 5, pp. 133β192, 1947 (Russian). View at Google Scholar - X. Y. Lin, βOscillation of second-order nonlinear neutral differential equations,β
*Journal of Mathematical Analysis and Applications*, vol. 309, no. 2, pp. 442β452, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH