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Abstract and Applied Analysis
Volume 2010, Article ID 428963, 16 pages
http://dx.doi.org/10.1155/2010/428963
Research Article

Nonoscillatory Solutions for Higher-Order Neutral Dynamic Equations on Time Scales

1College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China
2Department of Mathematics, Guangxi College of Finance and Economics, Nanning, Guangxi 530003, China

Received 16 May 2010; Revised 4 July 2010; Accepted 16 July 2010

Academic Editor: D. Anderson

Copyright © 2010 Taixiang Sun 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

We study the higher-order neutral dynamic equation for and obtain some necessary and sufficient conditions for the existence of nonoscillatory bounded solutions for this equation.

1. Introduction

A time scale is an arbitrary nonempty closed subset of the real numbers. Thus, , that is, the real numbers, the integers, and the natural numbers, are examples of time scales. We assume throughout that the time scale has the topology that it inherits from the real numbers with the standard topology.

The theory of time scale, which has recently received a lot of attention, was introduced by Hilger's landmark paper [1], a rapidly expanding body of literature has sought to unify, extend, and generalize ideas from discrete calculus, quantum calculus, and continuous calculus to arbitrary time scale calculus, where a time scale is a nonempty closed subset of the real numbers, and the cases when this time scale is equal to the real numbers or to the integers represent the classical theories of differential and difference equations. Many other interesting time scales exist, and they give rise to many applications (see [2]). Not only the new theory of the so-called “dynamic equations” unifies the theories of differential equations and difference equations but also extends these classical cases to cases “in between”, for example, to the so-called -difference equations when , which has important applications in quantum theory (see [3]).

On a time scale , the forward jump operator, the backward jump operator and the graininess function are defined as respectively. We refer the reader to [2, 4] for further results on time scale calculus.

In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions of various equations on time scales, and we refer the reader to the papers of [520].

In [21] Zhu and Wang studied the existence of nonoscillatory solutions to neutral dynamic equation

Karpuz [22] studied the asymptotic behavior of delay dynamic equations having the following form: Furthermore, Karpuz in [23] obtained necessary and sufficient conditions for the asymptotic behaviour of all bounded solutions of the following higher-order nonlinear forced neutral dynamic equation and also studied in [24] oscillation of unbounded solutions of a similar type of equations.

Li et al. [25] considered the existence of nonoscillatory solutions to the second-order neutral delay dynamic equation of the form

In [26, 27], Zhang et al. obtained some sufficient conditions for the existence of nonoscillatory solutions for the following higher-order equations:

respectively.

Motivated by the above studies, in this paper, we investigate the existence of nonoscillatory solutions of the following higher order neutral dynamic equation: where , is the quotient of odd positive integers, , the time scale interval , , , with and satisfying the following conditions:(i) for any and .(ii) is nondecreasing in for any .

Since we are interested in the oscillatory behavior of solutions near infinity, we assume that . By a solution of (1.7) we mean a nontrivial real-valued function , such that and satisfies (1.7) on . The solutions vanishing in some neighborhood of infinity will be excluded from our consideration. A solution of (1.7) is said to be oscillatory if it is neither eventually positive nor eventually negative, otherwise it is called nonoscillatory.

2. Auxiliary Results

We state the following conditions, which are needed in the sequel:

;

there exist constants with such that for all

there exist constants such that for all ;

there exist constants such that for all .

Let be a nonnegative integer and ; we define two sequences of functions and as follows:

By Theorems and of [2], we have

where and denote for each fixed the derivatisve of and with respect to , respectively.

Lemma 2.1 (see [23, 24]). Assume that and , then

Lemma 2.2 (see [23, 24]). Let be a nonnegative integer, , and . Then each of the following is true:(i) implies that for all ;(ii) implies that for all .

Lemma 2.3 (see [23]). Let be a nonnegative integer, and . Then
implies that each of the following is true:
(i) is decreasing for all and all .(ii) for all .(iii) for all and all .

Lemma 2.4 (see [28]). Let and . Then(1) implies for all .(2) implies for all .

Lemma 2.5 (see [29]). Let be bounded for with , where . Then for and
In the sequel, write

Lemma 2.6. Assume that is bounded and holds. If is a bounded nonoscillatory solution of (1.7), then eventually.

Proof. Without loss of generality, assume that there is some such that and for . From (1.7) we have Thus, is strictly decreasing on . If there exists such that , then Therefore, we have By condition , we obtain . Then it follows from Lemma 2.4 that , which is a contradiction since and are bounded. Hence, for all . The proof is completed.
Let be the Banach space of all bounded rd-continuous functions on with sup norm . 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 , . is called completely continuous if it is continuous and maps bounded sets into relatively compact sets.

Lemma 2.7 (see [21]). Suppose that is bounded and uniformly Cauchy. Further, suppose that is equi-continuous on for any . Then is relatively compact.

Lemma 2.8 (see [21]). Suppose that is a Banach space and is a bounded, convex, and closed subset of . Suppose further that there exist two operators and such that(i) for all ,(ii) is a contraction mapping,(iii) is completely continuous. Then has a fixed point in .

3. Main Results and Examples

Now, we state and prove our main results.

Theorem 3.1. Assume that and hold. Then (1.7) has a nonoscillatory bounded solution with if and only if there exists some constant such that

Proof. Sufficiency. Assume that (1.7) has a nonoscillatory bounded solution on with . Without loss of generality, we assume that there is a constant and some such that and for . It follows from Lemma 2.6 that for . By assumption that is bounded and condition , we see that is bounded. Thus, by Lemma 2.5 we get that there exists such that Integrating (1.7) from to , we have Therefore, it follows from (3.2) and (3.3) that, for , By Lemma 2.2, we see that (3.1) holds.
Necessity. Suppose that there exists some constant such that where . Then by Lemma 2.3 there exists such that and for , where is a constant. Let It is easy to verify that is a bounded, convex, and closed subset of .
Now we define two operators and as follows: where for any and for any . Now we show that and satisfy the conditions in Lemma 2.8.
(1) We will show that for any . In fact, for any and , , which implies that for any .
(2) We will show that is a contraction mapping. Indeed, for any and , we have Therefore, we have which implies that is a contraction mapping.
(3) We will show that is a completely continuous mapping.
(i)By the proof of (1), we see that for . That is, . (ii)We consider the continuity of . Let and as , then and as for any . Consequently, for any , we have Since and, for any , we have By Chapter 5 in [4], we see that the Lebesgue dominated convergence theorem holds for the integral on time scales. Then which implies that is continuous on . (iii)We show that is uniformly Cauchy. In fact, for any , take so that Then for any and , we have which implies that is uniformly Cauchy. (iv)We show that is equicontinuous on for any . Without loss of generality, we assume . For any , choose , then when with , we have by Lemma 2.1 that for any , which implies that is equi-continuous on for any .
By Lemma 2.7, we see that is a completely continuous mapping. It follows from Lemma 2.8 that there exists such that , which is the desired bounded solution of (1.7) with . The proof is completed.

Theorem 3.2. Assume that and hold, and that has the inverse . Then (1.7) has a nonoscillatory bounded solution with if and only if there exists some constant such that (3.1) holds.

Proof. The proof of sufficiency is similar to that of Theorem 3.1.
Necessity. Suppose that there exists some constant such that where . Then by Lemma 2.3 there exists such that and for , where is a constant. Let It is easy to verify that is a bounded, convex and closed subset of .
Now we define two operators and as follows: where for any and for any . Now we show that and satisfy the conditions in Lemma 2.8.
We will show that for any . In fact, for any and , , and , which implies that for any and is uniformly bounded.
Now we show that is equicontinuous on for any . Without loss of generality, we assume that . Since are continuous on , so they are uniformly continuous on . For any , choose such that when with , we have Then, it follows from Lemma 2.1 that, for any , which implies that is equi-continuous on for any .
The rest of the proof is similar to that of Theorem 3.1. The proof is completed.

Theorem 3.3. Assume that and hold and that has the inverse . Then (1.7) has a nonoscillatory bounded solution with if and only if there exists some constant such that (3.1) holds.

Proof. The proof of sufficiency is similar to that of Theorem 3.1.
Necessity. Suppose that there exists some constant such that where . Then by Lemma 2.3 there exists such that and for , where is a constant. Let It is easy to verify that is a bounded, convex, and closed subset of .
Now we define two operators and as follows: where for any and for any . Now we show that and satisfy the conditions in Lemma 2.8.
We will show that for any . In fact, for any and , , and , which implies that for any and is uniformly bounded. The rest of the proof is similar to that of Theorem 3.2. The proof is completed.
According to the proofs of Theorem 3.1, Theorem 3.2, and Theorem 3.3 in [23], we have

Lemma 3.4. Assume that holds. Suppose that one of the following holds:(i) satisfies condition ,(ii) satisfies condition and has the inverse ,(iii) satisfies condition and has the inverse . If is a bounded nonoscillatory solution of (1.7), then implies that .
By Lemma 3.4, Theorem 3.1,Theorem 3.2 and Theorem 3.3, we have.

Corollary 3.5. Assume that holds. Suppose that one of the following holds:(i) satisfies condition (ii) satisfies condition and has the inverse ,(iii) satisfies condition and has the inverse . Then every bounded solution of (1.7) is oscillation or converges to zero at infinity if and only if there exists some constant such that

Example 3.6. Let with . Consider the following higher order dynamic equation: where , is the quotient of odd positive integers,,, where , , , and .
It is easy to verify that satisfies condition and . On the other hand, we have andfor any constant , That is, conditions and (3.1) hold. By Theorem 3.1, Theorem 3.2, and Theorem 3.3, we see that (3.33) has a nonoscillatory bounded solution with .

Remark 3.7. Results of this paper can be extended to the case with several delays easily.

Acknowledgment

Project Supported by NSF of China (10861002) and NSF of Guangxi (2010GXNSFA013106, 2011GXNSFA014781) and SF of Education Department of Guangxi (200911MS212) and Innovation Project of Guangxi Graduate Education 2010105930701M43.

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