# Existence of Nonoscillatory Solutions to Second-Order Neutral Delay Dynamic Equations on Time Scales

- Tongxing Li
^{1}, - Zhenlai Han
^{1, 2}Email author, - Shurong Sun
^{1, 3}and - Dianwu Yang
^{1}

**2009**:562329

https://doi.org/10.1155/2009/562329

© Tongxing Li et al. 2009

**Received: **5 March 2009

**Accepted: **24 August 2009

**Published: **11 October 2009

## Abstract

## 1. Introduction

The theory of time scales, which has recently received a lot of attention, was introduced by Hilger in his Ph.D. Thesis in 1988 in order to unify continuous and discrete analysis (see Hilger [1]). Several authors have expounded on various aspects of this new theory; see the survey paper by Agarwal et al. [2] and references cited therein. A book on the subject of time scales, by Bohner and Peterson [3], summarizes and organizes much of the time scale calculus; we refer also to the last book by Bohner and Peterson [4] for advances in dynamic equations on time scales. For the notation used below we refer to the next section that provides some basic facts on time scales extracted from Bohner and Peterson [3].

In recent years, there has been much research activity concerning the oscillation of solutions of various equations on time scales, and we refer the reader to Erbe [5], Saker [6], and Hassan [7]. And there are some results dealing with the oscillation of the solutions of second-order delay dynamic equations on time scales [8–22].

In this work, we will consider the existence of nonoscillatory solutions to the second-order neutral delay dynamic equation of the form

on a time scale (an arbitrary closed subset of the reals).

The motivation originates from Kulenović and Hadžiomerpahić [23] and Zhu and Wang [24]. In [23], the authors established some sufficient conditions for the existence of positive solutions of the delay equation

Recently, [24] established the existence of nonoscillatory solutions to the neutral equation

Neutral equations find numerous applications in natural science and technology. For instance, they are frequently used for the study of distributed networks containing lossless transmission lines. So, we try to establish some sufficient conditions for the existence of equations of (1.1). However, there are few papers to discuss the existence of nonoscillatory solutions for neutral delay dynamic equations on time scales.

Since we are interested in the nonoscillatory behavior of (1.1), we assume throughout that the time scale under consideration satisfies and

As usual, by a solution of (1.1) we mean a continuous function which is defined on and satisfies (1.1) for A solution of (1.1) is said to be eventually positive (or eventually negative) if there exists such that (or ) for all in A solution of (1.1) is said to be nonoscillatory if it is either eventually positive or eventually negative; otherwise, it is oscillatory.

## 2. Main Results

In this section, we establish the existence of nonoscillatory solutions to (1.1). For let and Further, let denote all continuous functions mapping into and

Endowed on with the norm ( ) is a Banach space (see [24]). Let we say that is uniformly Cauchy if for any given there exists such that for any for all .

is said to be equicontinuous on if for any given there exists such that for any and with

Also, we need the following auxiliary results.

Lemma 2.1 (see [24, Lemma ]).

Suppose that is bounded and uniformly Cauchy. Further, suppose that is equicontinuous on for any Then is relatively compact.

Lemma 2.2 (see [25, Kranoselskii's fixed point theorem]).

- (i)
- (ii)
- (iii)

Throughout this section, we will assume in (1.1) that

, , = , = , , , , , = , and there exists a function such that = , =

Theorem 2.3.

Assume that holds and Then (1.1) has an eventually positive solution.

Proof.

Furthermore, from we see that there exists with such that for

It is easy to verify that is a bounded, convex, and closed subset of

Next, we will show that and satisfy the conditions in Lemma 2.2.

for any Hence, is a contraction mapping.

We will prove that is a completely continuous mapping. First, by we know that maps into

which proves that is continuous on

for all This means that is equicontinuous on for any

By means of Lemma 2.1, is relatively compact. From the above, we have proved that is a completely continuous mapping.

which implies that is an eventually positive solution of (1.1). The proof is complete.

Theorem 2.4.

Assume that holds and Then (1.1) has an eventually positive solution.

Proof.

Furthermore, from we see that there exists with such that for

It is easy to verify that is a bounded, convex, and closed subset of

Now we define two operators and as in Theorem 2.3 with replaced by The rest of the proof is similar to that of Theorem 2.3 and hence omitted. The proof is complete.

Theorem 2.5.

Assume that holds and Then (1.1) has an eventually positive solution.

Proof.

Furthermore, from we see that there exists with such that for

It is easy to verify that is a bounded, convex, and closed subset of

Now we define two operators and as in Theorem 2.3 with replaced by The rest of the proof is similar to that of Theorem 2.3 and hence omitted. The proof is complete.

We will give the following example to illustrate our main results.

Example 2.6.

where , , , , , , , Then , , Let It is easy to see that the assumption holds. By Theorem 2.3, (2.26) has an eventually positive solution.

## Declarations

### Acknowledgments

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have lead to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (60774004), China Postdoctoral Science Foundation Funded Project (20080441126), Shandong Postdoctoral Funded Project (200802018), Shandong Research Funds (Y2008A28, Y2007A27), and also supported by the University of Jinan Research Funds for Doctors (B0621, XBS0843).

## Authors’ Affiliations

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