Abstract and Applied Analysis

Volume 2013 (2013), Article ID 740568, 11 pages

http://dx.doi.org/10.1155/2013/740568

Research Article

Oscillation Criteria for Fourth-Order Nonlinear Dynamic Equations on Time Scales

Xin Wu,¹ Taixiang Sun,¹ Hongjian Xi,² and Changhong Chen¹

¹College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China

²Department of Mathematics, Guangxi College of Finance and Economics, Nanning, Guangxi 530003, China

Received 6 May 2013; Revised 20 June 2013; Accepted 21 June 2013

Academic Editor: Delfim F. M. Torres

Copyright © 2013 Xin Wu et al. 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.

Abstract.

We establish some new oscillation criteria for nonlinear dynamic equation of the form on an arbitrary time scale with , where are positive rd-continuous functions. An example illustrating the importance of our result is included.

1. Introduction

A time scaleis an arbitrary nonempty closed set of real numberswith the topology and ordering inherited from. The theory of time scales, which has recently received a lot of attention, was introduced by Hilger in his Ph.D thesis [1] in order to unify continuous and discrete analysis. The cases when a time scaleis equal toor the set of all integersrepresent the classical theories of differential and difference equations. Many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different from their continuous counterparts. The study of dynamic equations on time scales reveals such discrepancies and helps avoid proving results twice once for differential equations and once again for difference equations. The general is to prove a result for a dynamic equation where the domain of the unknown function is a time scale. In this way results not only related to the set of real numbers or set of integers but those pertaining to more general time scales are obtained. Therefore, not only can the theory of dynamic equations unify 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 [2]). In the last years there has been much research activity concerning the oscillation and asymptotic behavior of solutions of some dynamic equations on time scales, and we refer the reader to the paper [3–8] and the references cited therein.

Recently, Hassan in [9] studied the third-order dynamic equation

on a time scale, whereis the quotient of odd positive integers,andare positive rd-continuous functions on, and the so-called delay functionsatisfiesforandandand obtained some oscillation criteria, which improved and extended the results that have been established in [10–12].

Li et al. in [13] also discussed the oscillation of (1), whereis the quotient of odd positive integers,is assumed to satisfyfor, and there exists a positive rd-continuous functionon such thatfor. They established some new sufficient conditions for the oscillation of (1).

Wang and Xu in [14] extended the Hille and Nehari oscillation theorems to the third-order dynamic equation

on a time scale, whereis a ratio of odd positive integers and the functionsare positive real-valued rd-continuous functions defined on.

Erbe et al. in [15] were concerned with the oscillation of the third-order nonlinear functional dynamic equation

on a time scale, whereis the quotient of odd positive integers,andare positive rd-continuous functions on, andsatisfiesand. The authors obtain some new oscillation criteria and extend many known results for oscillation of third-order dynamic equations.

Qi and Yu in [16] obtained some oscillation criteria for the fourth-order nonlinear delay dynamic equation

on a time scale, whereis the ratio of odd positive integers,is a positive real-valued rd-continuous function defined on,, and.

Grace et al. in [17] were concerned with the oscillation of the fourth-order nonlinear dynamic equation

on a time scale, whereis the ratio of odd positive integers,is a positive real-valued rd-continuous function defined on. They reduce the problem of the oscillation of all solutions of (5) to the problem of oscillation of two second-order dynamic equations and give some conditions ensuring that all bounded solutions of (5) are oscillatory.

Grace et al. in [18] establish some new criteria for the oscillation of fourth-order nonlinear dynamic equations

whereis a positive real-valued rd-continuous function satisfying that,is continuous satisfyingandforand. They also investigate the case of strongly superlinear and the case of strongly sublinear equations subject to various conditions.

Agarwal et al. in [19] were concerned with oscillatory behavior of a fourth-order half-linear delay dynamic equation with damping

on a time scalewith, whereis the ratio of odd positive integers,are positive real-valued rd-continuous functions defined on, andas. They establish some new oscillation criteria of (7).

Zhang et al. in [20] were concerned with the oscillation of a fourth-order nonlinear dynamic equation

on an arbitrary time scalewith  , wherewithand there exists a positive constantsuch thatfor all; they give a new oscillation result of (8).

Motivated by the previous studies, in this paper, we will study the oscillation criteria of the following fourth-order nonlinear dynamic equation:

whereis a time scale with  is a constant and. Throughout this paper, we assume that the following conditions are satisfied:

(H ₁) and .(H ₂)(H ₃)and there exists a positive constantsuch that for any,.

By a solution of (9), we mean a nontrivial real-valued functionwith, which has the property thatand satisfies (9) on, whereis the space of differentiable functions whose derivative is rd-continuous. The solutions vanishing in some neighborhood of infinity will be excluded from our consideration. A solutionof (9) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise it is called nonoscillatory.

2. Some Auxiliary Lemmas

We shall employ the following lemmas.

Lemma 1. Assume thatis an eventually positive solution of (9). Then there existssufficiently large, such that, for, one of the following cases holds:

(1), , , ,(2), , , ,(3), , , ,(4), , , .

Proof. Letbe an eventually positive solution of (9). Then there is a, sufficiently large, such that,for. By (9) we have

which implies that is decreasing and one of the following two cases holds.

(a)for.(b) There is asuch that.

If case (a) holds, thenis strictly increasing onand there exist the following two subcases.

()for.() There exists asuch that.

If subcaseholds, then we claim. If not, there exists asuch thatfor. Thus, we get

which contradicts eventually. Therefore, we obtain case (4).

If subcaseholds, then let we get

Therefore, we obtain case (3).

If case (b) holds, then we claimfor. If not, there exists asuch thatfor. Integrating this inequality fromto, we get

Then, there exists a such thatfor. Integrating this inequality fromto, we get

which contradicts eventually. The proof is completed.

Lemma 2 (see [12]). Assume that there existssuch thatsatisfies

Then

where.

3. The Main Result

Now we state and prove our main result.

Theorem 3. Assume that one of the following conditions holds:

If there exist two positive functionssuch that for all sufficiently large, and, and some constant,

where

Then, every solutionof (9) is oscillatory.

Proof. Assume that (9) has a nonoscillatory solutionon. Then, without loss of generality, there is a, sufficiently large, such thatfor. By Lemma 1, there exist the following four possible cases:

(1), , ,(2), , ,(3), , ,(4), , .

If case (1) holds, then

which implies thatis decreasing on, and so

Dividing the previous inequality byand integrating the resulting inequality fromto, we get

Let, we obtain

Hence, there exists a constantsuch that

Integrating (28) fromto, we get

which implies that

which contradicts assumption (17).

Integrating (28) fromto, we get

Integrating the previous inequality fromtogives

which implies

which contradicts assumption (18).

Let. Integrating (27) fromtogives

Integrating (34) fromto, we get

Set

Then,forand

By (34), we get

Combining (36) with (38) gives

In view of (35), we get

From (39), we obtain

Integrating (41) fromtogives

which implies

Which contradicts assumption (19).

If case (2) holds, then set

andforand

On the other hand, letfor, where; it is easy to check thatIn view of

by, we get

Therefore, by Lemma 2, for any, there existssuch that

Then, we see that

Since

we get

In view of (49), we obtain that for all,

On the other hand, there existssuch that for any,

It follows from (52) and (53) that

Combining (45) with (54) gives

By (27) we get

Multiplying both sides of (55) withreplaced by, by, and integrating with respect tofromto, one gets

Thus,

which implies that

which contradicts assumption (20).

If case (3) holds, then since

we have

Hence, there existssuch that

which implies that

Hence, there existssuch that

Combining (62) with (64) gives

Write

Thus,and for any,

By (61) and (65), we get

Integrating the last inequality fromto, we get

which contradicts assumption (21).

If case (4) holds, then

Integrating the previous inequality fromto, we get

Lettingin this inequality, we obtain

Integrating the previous inequality fromto, we get

Lettingin this inequality, we obtain

Now we set

Thus,and for any ,

Since

we get

Hence, by (74) and (78), we get

Integrating the previous inequality fromto, we get

which contradicts assumption (22). The proof is completed.

4. Example

Finally, we give an example to illustrate our main result.

Example 1. Consider the fourth-order nonlinear dynamic equation

whereis a constant, and

So. It is easy to calculate that

It is obvious that

Therefore, we get

Then, condition (18) holds. By Lemma 2, we get

whilesufficiently large. Let. We have that if, then

Since

we get

Since

we obtain

Hence, conditions (18), (20), (21), and (22) of Theorem 3 are satisfied. By Theorem 3, we see that every solutionof (81) is oscillatory if.

Acknowledgments

This project is supported by NNSF of China (11261005) and NSF of Guangxi (2011GXNSFA018135, 2012GXNSFD A276040) and SF of ED of Guangxi (2013ZD061).

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