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Asymptotic Behavior of Solutions of HigherOrder Dynamic Equations on Time Scales
Advances in Difference Equations volumeÂ 2011, ArticleÂ number:Â 237219 (2011)
Abstract
We investigate the asymptotic behavior of solutions of the following higherorder 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
In this paper, we investigate the asymptotic behavior of solutions of the following higherorder dynamic equation
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 realvalued function , which has the property that and satisfies (1.1) on , where is the space of rdcontinuous 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 socalled "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 socalled 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. Let with , for all , then the delta exponential function 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 [5â€“18].
Recently, Erbe et al. [19â€“21] considered the asymptotic behavior of solutions of the thirdorder dynamic equations
respectively, and established some sufficient conditions for oscillation.
Karpuz [22] studied the asymptotic nature of all bounded solutions of the following higherorder nonlinear forced neutral dynamic equation
Chen [23] derived some sufficient conditions for the oscillation and asymptotic behavior of the thorder 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 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
Let be a nonnegative integer and , then we define a sequence of functions as follows:
To obtain our main results, we need the following lemmas.
Lemma 2.1.
Let be a positive integer, then there exists , such that
Proof.
We will prove the above by induction. First, if , then we take . Thus,
Next, we assume that there exists , such that for and with , then
from which it follows that there exists , such that for and . The proof is completed.
Lemma 2.2 (see [24]).
Let , then
Lemma 2.3 (see [2]).
Let and , then
implies
Lemma 2.4 (see [2]).
Let be a positive integer. Suppose that is times differentiable on T. Let and , then
Lemma 2.5 (see [2]).
Assume that and are differentiable on T with . If there exists , such that
then
Lemma 2.6 (see [23]).
Let be defined on , and with for and not eventually zero. If is bounded, then

(1)
for ,

(2)
for all and .
Now, one states and proves the main results.
Theorem 2.7.
Assume that there exists , such that the function satisfies
where are nonnegative functions on and
with , then every solution of (1.1) satisfies one of the following conditions:

(1)
,

(2)
there exist constants with , such that
(2.13)
Proof.
Let be a solution of (1.1), then it follows from Lemma 2.4 that for ,
By (2.11) and Lemma 2.1, we see that there exists , such that for and ,
Then we obtain
where
with
Using (2.16) and (2.17), it follows that
By Lemma 2.3, we have
with . Hence from (2.12), there exists a finite constant , such that for . Thus, inequality (2.20) implies that
By (1.1), we see that if , 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 . Therefore, exists and is a finite number. Let . If , then it follows from Lemma 2.5 that
and has the desired asymptotic property. The proof is completed.
Theorem 2.8.
Assume that there exist functions , and nondecreasing continuous functions , and such that
with
then every solution of (1.1) satisfies one of the following conditions:

(1)
,

(2)
there exist constants with such that
Proof.
Let be a solution of (1.1), then it follows from Lemma 2.4 that for ,
By Lemma 2.1 and (2.25), we see that there exists , such that for and ,
Then, we obtain
where
with
Using (2.30) and (2.31), it follows that
Write
then
from which it follows that
Since and is strictly increasing, there exists a constant , such that for . By (2.30), (2.33), and (2.34), we have
It follows from (1.1) that if , then
Since (2.38) and condition (2.25) implies that
we see that the sum in (2.39) converges as . Therefore, exists and is a finite number. Let . If , then it follows from Lemma 2.5 that
and has the desired asymptotic property. The proof is completed.
Theorem 2.9.
Assume that there exist positive functions , and nondecreasing continuous functions , and , such that
with
then every solution of (1.1) satisfies one of the following conditions:

(1)
,

(2)
there exist constants with , such that
Proof.
Arguing as in the proof of Theorem 2.8, we see that there exists , such that for and ,
from which we obtain
where
Using (2.46) and (2.47), it follows that
Write
then
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.
Assume that the function satisfies

(1)
for all ,

(2)
for and ,

(3)
for and is continuous at with ,
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.
Assume that (1.1) has a nonoscillatory solution on , then, without loss of generality, there is a , sufficiently large, such that for . It follows from (1.1) that for and not eventually zero. By Lemma 2.6, we have
and is eventually monotone. Also for if is even and for if is odd. Since is bounded, we find . Furthermore, if is even, then .
We claim that . If not, then there exists , such that
since is continuous at by the condition (3). From (1.1) and (2.55), we have
Multiplying the above inequality by , and integrating from to , we obtain
Since
we get
where . Thus, since is bounded, which gives a contradiction to the condition (2). The proof is completed.
3. Examples
Example 3.1.
Consider the following higherorder dynamic equation:
where and . Let and
then we have
by Exampleâ€‰â€‰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.
Consider the following higherorder dynamic equation:
where , , and . Let , , and
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.
Consider the following higherorder dynamic equation:
where with and . Let , and
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 .
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Acknowledgment
This paper was supported by NSFC (no. 10861002) and NSFG (no. 2010GXNSFA013106, no. 2011GXNSFA018135) and IPGGE (no. 105931003060).
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Sun, T., Xi, H. & Peng, X. Asymptotic Behavior of Solutions of HigherOrder Dynamic Equations on Time Scales. Adv Differ Equ 2011, 237219 (2011). https://doi.org/10.1155/2011/237219
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DOI: https://doi.org/10.1155/2011/237219
Keywords
 Asymptotic Behavior
 Dynamic Equation
 Difference Equation
 Nonoscillatory Solution
 Jump Operator