1. Introduction

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

(1.1)

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

(1.2)

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

(1.3)

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].

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

(1.4)

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

(1.5)

Chen [23] derived some sufficient conditions for the oscillation and asymptotic behavior of the th-order nonlinear neutral delay dynamic equations

(1.6)

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:

(2.1)

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

Lemma 2.1.

Let be a positive integer, then there exists , such that

(2.2)

Proof.

We will prove the above by induction. First, if , then we take . Thus,

(2.3)

Next, we assume that there exists , such that for and with , then

(2.4)

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

Lemma 2.2 (see [24]).

Let , then

(2.5)

Lemma 2.3 (see [2]).

Let and , then

(2.6)

implies

(2.7)

Lemma 2.4 (see [2]).

Let be a positive integer. Suppose that is times differentiable on T. Let and , then

(2.8)

Lemma 2.5 (see [2]).

Assume that and are differentiable on T with . If there exists , such that

(2.9)

then

(2.10)

Lemma 2.6 (see [23]).

Let be defined on , and with for and not eventually zero. If is bounded, then

  1. (1)

    for ,

  2. (2)

    for all and .

Now, one states and proves the main results.

Theorem 2.7.

Assume that there exists , such that the function satisfies

(2.11)

where are nonnegative functions on and

(2.12)

with , then every solution of (1.1) satisfies one of the following conditions:

  1. (1)

    ,

  2. (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 ,

(2.14)

By (2.11) and Lemma 2.1, we see that there exists , such that for and ,

(2.15)

Then we obtain

(2.16)

where

(2.17)

with

(2.18)

Using (2.16) and (2.17), it follows that

(2.19)

By Lemma 2.3, we have

(2.20)

with . Hence from (2.12), there exists a finite constant , such that for . Thus, inequality (2.20) implies that

(2.21)

By (1.1), we see that if , then

(2.22)

Since condition (2.12) and Lemma 2.2 implies that

(2.23)

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

(2.24)

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

(2.25)

with

(2.26)

then every solution of (1.1) satisfies one of the following conditions:

  1. (1)

    ,

  2. (2)

    there exist constants with such that

(2.27)

Proof.

Let be a solution of (1.1), then it follows from Lemma 2.4 that for ,

(2.28)

By Lemma 2.1 and (2.25), we see that there exists , such that for and ,

(2.29)

Then, we obtain

(2.30)

where

(2.31)

with

(2.32)

Using (2.30) and (2.31), it follows that

(2.33)

Write

(2.34)
(2.35)

then

(2.36)

from which it follows that

(2.37)

Since and is strictly increasing, there exists a constant , such that for . By (2.30), (2.33), and (2.34), we have

(2.38)

It follows from (1.1) that if , then

(2.39)

Since (2.38) and condition (2.25) implies that

(2.40)

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

(2.41)

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

(2.42)

with

(2.43)

then every solution of (1.1) satisfies one of the following conditions:

  1. (1)

    ,

  2. (2)

    there exist constants with , such that

(2.44)

Proof.

Arguing as in the proof of Theorem 2.8, we see that there exists , such that for and ,

(2.45)

from which we obtain

(2.46)

where

(2.47)
(2.48)

Using (2.46) and (2.47), it follows that

(2.49)

Write

(2.50)
(2.51)

then

(2.52)

from which it follows that

(2.53)

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. (1)

    for all ,

  2. (2)

    for and ,

  3. (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

(2.54)

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

(2.55)

since is continuous at by the condition (3). From (1.1) and (2.55), we have

(2.56)

Multiplying the above inequality by , and integrating from to , we obtain

(2.57)

Since

(2.58)

we get

(2.59)

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:

(3.1)

where and . Let and

(3.2)

then we have

(3.3)

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:

(3.4)

where , , and . Let , , and

(3.5)

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:

(3.6)

where with and . Let , and

(3.7)

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 .