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

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## Abstract

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

## Keywords

Asymptotic Behavior Dynamic Equation Difference Equation Nonoscillatory Solution Jump Operator## 1. Introduction

on an arbitrary time scale Open image in new window , where the function Open image in new window is defined on Open image in new window .

Since we are interested in the asymptotic and oscillatory behavior of solutions near infinity, we assume that Open image in new window , and define the time scale interval Open image in new window , where Open image in new window . By a solution of (1.1), we mean a nontrivial real-valued function Open image in new window , which has the property that Open image in new window and satisfies (1.1) on Open image in new window , where Open image in new window is the space of rd-continuous functions. The solutions vanishing in some neighborhood of infinity will be excluded from our consideration. A solution Open image in new window 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 Open image in new window -difference equations when Open image in new window , 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, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 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 Open image in new window of (1.1) satisfies one of the following conditions: (1) Open image in new window ; (2) there exist constants Open image in new window with Open image in new window , such that Open image in new window , where Open image in new window 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 Open image in new window , such that Open image in new window for Open image in new window and Open image in new window . The proof is completed.

Lemma 2.2 (see [24]).

Lemma 2.3 (see [2]).

Lemma 2.4 (see [2]).

**T**. Let Open image in new window and Open image in new window , then

Lemma 2.5 (see [2]).

**T**with Open image in new window . If there exists Open image in new window , such that

Lemma 2.6 (see [23]).

- (1)
- (2)

Now, one states and proves the main results.

Theorem 2.7.

- (1)
- (2)
there exist constants Open image in new window with Open image in new window , such that

(2.13)

Proof.

and Open image in new window has the desired asymptotic property. The proof is completed.

Theorem 2.8.

- (1)
- (2)
there exist constants Open image in new window with Open image in new window such that

Proof.

and Open image in new window has the desired asymptotic property. The proof is completed.

Theorem 2.9.

- (1)
- (2)
there exist constants Open image in new window with Open image in new window , such that

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.

- (1)
- (2)
- (3)
Open image in new window for Open image in new window and Open image in new window is continuous at Open image in new window with Open image in new window ,

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

Proof.

and Open image in new window is eventually monotone. Also Open image in new window for Open image in new window if Open image in new window is even and Open image in new window for Open image in new window if Open image in new window is odd. Since Open image in new window is bounded, we find Open image in new window . Furthermore, if Open image in new window is even, then Open image in new window .

where Open image in new window . Thus, Open image in new window since Open image in new window 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 Open image in new window is a solution of (3.1) with Open image in new window , then there exist constants Open image in new window with Open image in new window , such that Open image in new window .

Example 3.2.

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

Example 3.3.

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

## Notes

### Acknowledgment

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

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