Discrete Dynamics in Nature and Society

Volume 2008, Article ID 150163, 9 pages

http://dx.doi.org/10.1155/2008/150163

## On the Nonoscillation of Second-Order Neutral Delay Differential Equation with Forcing Term

^{1}School of Mechantronic Engineering, North University of China, Taiyuan 030051, China^{2}Department of Basic Science, Taiyuan Institute of Technology, Taiyuan 030008, China^{3}Department of Mathematics, North University of China, Taiyuan 030051, China^{4}Department of Mathematics, Shanxi University, Taiyuan 030006, China

Received 3 September 2008; Accepted 19 November 2008

Academic Editor: Guang Zhang

Copyright © 2008 Jin-Zhu Zhang 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 nonoscillation of second-order neutral delay differential equation with forcing term. By using contraction mapping principle, some sufficient conditions for the existence of nonoscillatory solution are established. The criteria obtained in this paper complement and extend several known results in the literature. Some examples illustrating our main results are given.

#### 1. Introduction

During the last two decades, there has been much research activity concerning the oscillation and nonoscillation of solutions of neutral type delay differential equations (see [1–9]). Investigation of such equations or systems, besides of their theoretical interest, have some importance in modelling of the networks containing lossless transmission lines, in the study of vibrating masses attached to an elastic bar and also in population dynamics, and so forth (see [1, 2, 8, 10] and the references cited therein).

In this paper, we consider the second-order neutral delay differential equation with forcing term of the formwhere

Let , where , be a given function and let be a given constant. By the method of steps (see [2]), it is easy to know that (1.1) has a unique solution in the sense that is twice continuously differential for , satisfies (1.1) and

As is customary, a solution of (1.1) is called oscillatory if it has arbitrarily large zeros, and otherwise it is nonoscillatory. Equation (1.1) is oscillatory if all its solutions are oscillatory.

When and the forcing term , (1.1) reduces to where and . The first global result of (1.4) (with respect to ), which is a sufficient condition for the existence of a nonoscillatory solution for all values of , have been examined by Kulenović and Hadžiomerspahić [4].

Recently, Parhi and Rath [7] studied oscillation behaviors for forced first-order neutral differential equations as follows Necessary and sufficient conditions are obtained in various ranges for so that every solution of (1.5) is oscillatory or tends to zero or to as .

Motivated by the idea of [4, 7], in present paper we establish sufficient conditions for existence of a nonoscillatory solution to (1.1) depending on various ranges of . Hereinafter, we assume that the following conditions hold,

(H1) and .(H2)There exists a function such that and .

#### 2. Main results

Theorem 2.1. *Suppose
that conditions and hold. If is in one of the following ranges: ** then (1.1) has a
nonoscillatory solution.*

*Proof. *The proof of
this theorem will be divided into four cases in terms of the four different
ranges of .*Case (i) (). *Choose a sufficiently large such that where and are positive constants such that

Let be the set of all continuous and bounded
functions on with the sup norm. SetDefine a mapping as follows:

Clearly, is continuous. For every and ,
using (2.3) and (2.5),
we get Furthermore, from (2.4) and (2.5),
we haveThus we prove that .
To apply the contraction principle, we need to prove that is a contraction mapping on since is a bounded, closed, and convex subset of .

Now, for and we havewhere we used sup norm. From (2.2),
we obtain ,
which completes the proof of Case (i).*Example 2.2. *Consider the second-order neutral delay
differential equationwhere such that Since as , ,
and ,
then the sufficient conditions—in Case (i) of Theorem 2.1—are satisfied.
Therefore, the equation has a positive solution. In fact is a positive solution of this equation.*Case (ii) (). *Choose a sufficiently large such that where and are positive constants such that

Let be the set as in Case (i). Set Define a mapping as follows:

Clearly, is continuous. For every and ,
using (2.14) and (2.16),
we getFurthermore, from (2.15) and (2.16) we haveThus we prove that .
To apply the contraction principle, we need to prove that is a contraction mapping on since is a bounded, closed, and convex subset of .

Now, for and ,
we havewhere we used sup norm. From (2.13),
we obtain ,
which completes the proof of Case (ii).*Example 2.3. *Consider the second-order neutral delay
differential equationwhere such that Since as , ,
and ,
then the sufficient conditions—in Case (ii) of Theorem 2.1—are satisfied.
Therefore, the equation has a positive solution. In fact is a positive solution of this equation.*Case (iii) (). *Choose a sufficiently large such that where and are positive constants such that

Let be the set as in Case (i). Set Define a mapping as follows:

Clearly, is continuous. For every and ,
using (2.25) and (2.27),
we getFurthermore, from (2.26) and (2.27),
we haveThus we prove that .
To apply the contraction principle, we need to prove that is a contraction mapping on since is a bounded, closed, and convex subset of .

Now, for and ,
we havewhere we used sup norm. From (2.24),
we obtain ,
which completes the proof of Case (iii).*Example 2.4. *Consider the second-order neutral delay
differential equation where .
This equation has a nonoscillatory solution since the sufficient conditions—in Case
(iii) of Theorem 2.1—are satisfied.*Case (iv) (). *Choose a sufficiently large such thatwhere and are positive constants such that

Let be the set as in Case (i). SetDefine a mapping as follows

Clearly, is continuous. For every and ,
using (2.37) and (2.38),
we getFurthermore, from
(2.36) and (2.38),
we haveThus we prove that .
To apply the contraction principle, we need to prove that is a contraction mapping on since is a bounded, closed, and convex subset of .

Now, for and ,
we havewhere we used sup norm. From (2.35),
we obtain ,
which completes the proof of Case (iv).*Example 2.5. *Consider the second-order neutral delay
differential equationwhere .
This equation has a nonoscillatory solution since the sufficient conditions—in Case (iv)
of Theorem 2.1—are satisfied.

#### Acknowledgments

This work was supported by Natural Science Foundations of Shanxi Province (2007011019), by the special Scientific Research Foundation for the subject of doctor in university (20060110005), and by Program for New Century Excellent Talents in University (NCET050271). The authors are very grateful to the referees for their useful comments.

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