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Asymptotic Behavior of Solutions of Higher-Order Dynamic Equations on Time Scales
Advances in Difference Equations volume 2011, Article number: 237219 (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
In this paper, 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
.
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]).
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 [5–18].
Recently, Erbe et al. [19–21] 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 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 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 higher-order dynamic equation:

where and
. Let
and

then we have

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.
Consider the following higher-order 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 higher-order 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 Higher-Order 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