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Advances in Difference Equations

, 2011:237219 | Cite as

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

Open Access
Research Article

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

In this paper, we investigate the asymptotic behavior of solutions of the following higher-order dynamic equation

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]).

On a time scale Open image in new window , 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. Let Open image in new window with Open image in new window , for all Open image in new window , then the delta exponential function Open image in new window is defined as the unique solution of the initial value problem

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].

Recently, Erbe et al. [19, 20, 21] considered the asymptotic behavior of solutions of the third-order dynamic equations

respectively, and established some sufficient conditions for oscillation.

Karpuz [22] studied the asymptotic nature of all bounded solutions of the following higher-order nonlinear forced neutral dynamic equation
Chen [23] derived some sufficient conditions for the oscillation and asymptotic behavior of the Open image in new window th-order nonlinear neutral delay dynamic equations

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

Let Open image in new window be a nonnegative integer and Open image in new window , then we define a sequence of functions Open image in new window as follows:

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

Lemma 2.1.

Let Open image in new window be a positive integer, then there exists Open image in new window , such that

Proof.

We will prove the above by induction. First, if Open image in new window , then we take Open image in new window . Thus,

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]).

Let Open image in new window be a positive integer. Suppose that Open image in new window is Open image in new window times differentiable on T. Let Open image in new window and Open image in new window , then

Lemma 2.5 (see [2]).

Assume that Open image in new window and Open image in new window are differentiable on T with Open image in new window . If there exists Open image in new window , such that

Lemma 2.6 (see [23]).

Now, one states and proves the main results.

Theorem 2.7.

Assume that there exists Open image in new window , such that the function Open image in new window satisfies
where Open image in new window are nonnegative functions on Open image in new window and
with Open image in new window , then every solution Open image in new window of (1.1) satisfies one of the following conditions:
  1. (1)
     
  2. (2)

    there exist constants Open image in new window with Open image in new window , such that

     

Proof.

Let Open image in new window be a solution of (1.1), then it follows from Lemma 2.4 that for Open image in new window ,
By (2.11) and Lemma 2.1, we see that there exists Open image in new window , such that for Open image in new window and Open image in new window ,
Then we obtain
Using (2.16) and (2.17), it follows that
By Lemma 2.3, we have
with Open image in new window . Hence from (2.12), there exists a finite constant Open image in new window , such that Open image in new window for Open image in new window . Thus, inequality (2.20) implies that
By (1.1), we see that if Open image in new window , then
Since condition (2.12) and Lemma 2.2 implies that
we find from (2.11) and (2.21) that the sum in (2.22) converges as Open image in new window . Therefore, Open image in new window exists and is a finite number. Let Open image in new window . If Open image in new window , then it follows from Lemma 2.5 that

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

Theorem 2.8.

Assume that there exist functions Open image in new window , and nondecreasing continuous functions Open image in new window , and Open image in new window such that
then every solution Open image in new window of (1.1) satisfies one of the following conditions:
  1. (1)
     
  2. (2)

    there exist constants Open image in new window with Open image in new window such that

     

Proof.

Let Open image in new window be a solution of (1.1), then it follows from Lemma 2.4 that for Open image in new window ,
By Lemma 2.1 and (2.25), we see that there exists Open image in new window , such that for Open image in new window and Open image in new window ,
Then, we obtain
Using (2.30) and (2.31), it follows that
from which it follows that
Since Open image in new window and Open image in new window is strictly increasing, there exists a constant Open image in new window , such that Open image in new window for Open image in new window . By (2.30), (2.33), and (2.34), we have
It follows from (1.1) that if Open image in new window , then
Since (2.38) and condition (2.25) implies that
we see that the sum in (2.39) converges as Open image in new window . Therefore, Open image in new window exists and is a finite number. Let Open image in new window . If Open image in new window , then it follows from Lemma 2.5 that

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

Theorem 2.9.

Assume that there exist positive functions Open image in new window , and nondecreasing continuous functions Open image in new window , and Open image in new window , such that
then every solution Open image in new window of (1.1) satisfies one of the following conditions:
  1. (1)
     
  2. (2)

    there exist constants Open image in new window with Open image in new window , such that

     

Proof.

Arguing as in the proof of Theorem 2.8, we see that there exists Open image in new window , such that for Open image in new window and Open image in new window ,
from which we obtain
Using (2.46) and (2.47), it follows that
from which it follows that

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 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.

Assume that (1.1) has a nonoscillatory solution Open image in new window on Open image in new window , then, without loss of generality, there is a Open image in new window , sufficiently large, such that Open image in new window for Open image in new window . It follows from (1.1) that Open image in new window for Open image in new window and not eventually zero. By Lemma 2.6, we have

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 .

We claim that Open image in new window . If not, then there exists Open image in new window , such that
since Open image in new window is continuous at Open image in new window by the condition (3). From (1.1) and (2.55), we have
Multiplying the above inequality by Open image in new window , and integrating from Open image in new window to Open image in new window , we obtain

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.

Consider the following higher-order dynamic equation:
then we have

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.

Consider the following higher-order dynamic equation:

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.

Consider the following higher-order dynamic equation:

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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Copyright information

© Taixiang Sun et al. 2011

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceGuangxi UniversityNanningChina
  2. 2.Department of MathematicsGuangxi College of Finance and EconomicsNanningChina

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