# Asymptotic Behavior of Solutions of Higher-Order Dynamic Equations on Time Scales

- Taixiang Sun
^{1}Email author, - Hongjian Xi
^{2}and - Xiaofeng Peng
^{1}

**2011**:237219

https://doi.org/10.1155/2011/237219

© Taixiang Sun et al. 2011

**Received: **18 November 2010

**Accepted: **23 February 2011

**Published: **14 March 2011

## Abstract

We investigate the asymptotic behavior of solutions of the following higher-order dynamic equation , on an arbitrary time scale , where the function is defined on . We give sufficient conditions under which every solution of this equation satisfies one of the following conditions: (1) ; (2) there exist constants with , such that , where are as in Main Results.

## 1. Introduction

on an arbitrary time scale , where the function is defined on .

Since we are interested in the asymptotic and oscillatory behavior of solutions near infinity, we assume that , and define the time scale interval , where . By a solution of (1.1), we mean a nontrivial real-valued function , which has the property that and satisfies (1.1) on , where is the space of rd-continuous functions. The solutions vanishing in some neighborhood of infinity will be excluded from our consideration. A solution of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative, otherwise it is called nonoscillatory.

The theory of time scales, which has recently received a lot of attention, was introduced by Hilger's landmark paper [1] in order to create a theory that can unify continuous and discrete analysis. The cases when a time scale is equal to the real numbers or to the integers represent the classical theories of differential and of 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]).

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 [5–18].

respectively, and established some sufficient conditions for oscillation.

on an arbitrary time scale **T**. Motivated by the above studies, in this paper, we study (1.1) and give sufficient conditions under which every solution
of (1.1) satisfies one of the following conditions: (1)
; (2) there exist constants
with
, such that
, where
are as in Section 2.

## 2. Main Results

To obtain our main results, we need the following lemmas.

Lemma 2.1.

Proof.

from which it follows that there exists , such that for and . The proof is completed.

Lemma 2.2 (see [24]).

Lemma 2.3 (see [2]).

Lemma 2.4 (see [2]).

Lemma 2.5 (see [2]).

Lemma 2.6 (see [23]).

Now, one states and proves the main results.

Theorem 2.7.

Proof.

and has the desired asymptotic property. The proof is completed.

Theorem 2.8.

Proof.

and has the desired asymptotic property. The proof is completed.

Theorem 2.9.

Proof.

The rest of the proof is similar to that of Theorem 2.8, and the details are omitted. The proof is completed.

Theorem 2.10.

then (1) if is even, then every bounded solution of (1.1) is oscillatory; (2) if is odd, then every bounded solution of (1.1) is either oscillatory or tends monotonically to zero together with .

Proof.

and is eventually monotone. Also for if is even and for if is odd. Since is bounded, we find . Furthermore, if is even, then .

where . Thus, since is bounded, which gives a contradiction to the condition (2). The proof is completed.

## 3. Examples

Example 3.1.

by Example 5.60 in [4]. Thus, it follows from Theorem 2.7 that if is a solution of (3.1) with , then there exist constants with , such that .

Example 3.2.

It is easy to verify that satisfies the conditions of Theorem 2.8. Thus, it follows that if is a solution of (3.4) with , then there exist constants with , such that .

Example 3.3.

It is easy to verify that satisfies the conditions of Theorem 2.9. Thus, it follows that if is a solution of (3.6) with , then there exist constants with , such that .

## Declarations

### Acknowledgment

This paper was supported by NSFC (no. 10861002) and NSFG (no. 2010GXNSFA013106, no. 2011GXNSFA018135) and IPGGE (no. 105931003060).

## Authors’ Affiliations

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