Journal of Applied Mathematics

Volume 2011, Article ID 418136, 10 pages

http://dx.doi.org/10.1155/2011/418136

## Lyapunov-Type Inequalities for Some Quasilinear Dynamic System Involving the -Laplacian on Time Scales

^{1}College of Mathematics and Computer Science, Jishou University, Jishou 416000, Hunan, China^{2}College of Science, Hunan University of Technology, Zhuzhou 412007, Hunan, China

Received 30 June 2011; Accepted 25 August 2011

Academic Editor: Yansheng Liu

Copyright © 2011 Xiaofei He and Qi-Ming Zhang. 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 several new Lyapunov-type inequalities for some quasilinear dynamic system involving the -Laplacian on an arbitrary time scale , which generalize and improve some related existing results including the continuous and discrete cases.

#### 1. Introduction

In recent years, the theory of time scales (or measure chains) has been developed by several authors with one goal being the unified treatment of differential equations (the continuous case) and difference equations (the discrete case). A time scale is an arbitrary nonempty closed subset of the real numbers . We assume that is a time scale and has the topology that it inherits from the standard topology on the real numbers . The two most popular examples are and . In Section 2, we will briefly introduce the time scale calculus and some related basic concepts of Hilger [1–3]. For further details, we refer the reader to the books independently by Kaymakcalan et al. [4] and by Bohner and Peterson [5, 6].

Consider the following quasilinear dynamic system involving the -Laplaci-an on an arbitrary time scale :

It is obvious that system (1.1) covers the continuous quasilinear system and the corresponding discrete case, respectively, when and ; that is,

In 1907, Lyapunov [7] established the first so-called Lyapunov inequality if the Hill equation has a real solution such that Moreover the constant 4 in (1.3) cannot be replaced by a larger number, where is a piece-wise continuous and nonnegative function defined on .

It is a classical topic for us to study Lyapunov-type inequalities which have proved to be very useful in oscillation theory, disconjugacy, eigenvalue problems, and numerous other applications in the theory of differential and difference equations. So far, there are many literatures which improved and extended the classical Lyapunov including continuous and discrete cases. For example, inequality (1.3) has been generalized to discrete linear Hamiltonian system by Zhang and Tang [8], to second-order nonlinear differential equations by Eliason [9] and by Pachpatte [10], to second-order nonlinear difference system by He and Zhang [11], to the second-order delay differential equations by Eliason [12] and by Dahiya and Singh [13], to higher-order differential equations by Pachpatte [14], Yang [15, 16], Yang and Lo [17] and Cakmak and Tiryaki [18, 19]. Lyapunov-type inequalities for the Emden-Fowler-type equations can be found in Pachpatte [10], and for the half-linear equations can be found in Lee et al. [20] and Pinasco [21]. Recently, there has been much attention paid to Lyapunov-type inequalities for dynamic systems on time scales and some authors including Agarwal et al. [22], Jiang and Zhou [23], He [24], He et al. [25], Saker [26], Bohner et al. [27], and Ünal and Cakmak [28] have contributed the above results.

In this paper, we use the methods in [29] to establish some Lyapunov-type inequalities for system (1.1) on an arbitrary time scale .

#### 2. Preliminaries about the Time Scales Calculus

We introduce some basic notions connected with time scales.

*Definition 2.1 (see [6]). *Let . We define * the forward jump operator * by
while * the backward jump operator * by
In this definition, we put (i.e., if has a maximum ) and (i.e., if has a minimum ), where denotes the empty set. If , we say that is * right-scattered*, while if , we say that is * left-scattered*. Also, if and , then is called * right-dense*, and if and , then is called * left-dense*. Points that are right-scattered and left-scattered at the same time are called * isolated*. Points that are right-dense and left-dense at the same time are called * dense*. If has a left-scattered maximum , then we define otherwise; . The * graininess function * is defined by
We consider a function and define so-called delta (or Hilger) derivative of at a point .

*Definition 2.2 (see [6]). *Assume that is a function, and let . Then, we define to be the number (provided it exists) with the property that given any *,* there is a neighborhood of (i.e., for some ) such that
We call the delta (or Hilger) derivative of at .

Lemma 2.3 (see [6]). *Assume that are differential at , then, *(i)*for any constant and , the sum is differential at with *(ii)*if exists, then is continuous at ,*(iii)*if exists, then ,*(iv)*the product is differential at with *(v)*if , then is differential at and *

*Definition 2.4 (see [6]). *A function is called rd-continuous, provided it is continuous at right-dense points in and left-sided limits exist (finite) at left-dense points in and denotes by *. *

*Definition 2.5 (see [6]). *A function is called an antiderivative of , provided holds for all *.* We define the Cauchy integral by

The following lemma gives several elementary properties of the delta integral.

Lemma 2.6 (see [6]). *If and , then *(i)*,
*(ii)*,
*(iii)*,
*(iv)*,
*(v)*,
*(vi)*if , then *

The notation and will denote time scales intervals. For example, . To prove our results, we present the following lemma.

Lemma 2.7 (see [6]). * Let and with . For , one has
*

Lemma 2.8 (see [6]). * Let and with for . For , one has
*

#### 3. Lyapunov-Type Inequalities

Denote

First, we give the following hypothesis.(H1) and are rd-continuous real functions and for and . Furthermore, and satisfy for .

Theorem 3.1. * Let with . Suppose that hypothesis (H1) is satisfied. If (1.1) has a real solution satisfying the boundary value conditions
**
then one has
**
where and in what follows for .*

*Proof. * By (1.1) and Lemma 2.3(iv), we obtain
where . From Definition 2.5, integrating (3.5) from to , together with (3.3), we get
It follows from (3.1), (3.3), and Lemma 2.7 that
Similarly, it follows from (3.2), (3.3), and Lemma 2.7 that
From (3.7) and (3.8), we have
So, from (3.3), (3.6), (3.9), (H1), and Lemma 2.8, we have
where

Next, we prove that
If (3.12) is not true, there exist such that
From (3.6), (3.13), and Lemma 2.8, we have
It follows from the fact that that
Combining (3.7) with (3.15), we obtain that for , which contradicts (3.3). Therefore, (3.12) holds. From (3.10), (3.12), and (H1), we have
It follows from (3.11) and (3.16) that (3.4) holds.

Corollary 3.2. *Let with . Suppose that hypothesis (H1) is satisfied. If (1.1) has a real solution satisfying the boundary value conditions (3.3), then one has
*

*Proof. * Since
it follows from (3.4) and (H1) that (3.17) holds.

Corollary 3.3. * Let with . Suppose that hypothesis (H1) is satisfied. If (1.1) has a real solution satisfying the boundary value conditions (3.3), then one has
**
where . *

*Proof. * Since
it follows from (3.20) and (H1) that (3.19) holds.

When , , and , system (1.1) reduces to a second-order half-linear dynamic equation, and denote by

We can easily derive the following corollary for (3.21).

Corollary 3.4. *Let with . If (3.21) has a solution satisfying
**
then
*

Especially, while , , and , system (1.1) reduces to a second-order linear dynamic equation and denote by

Obviously, (3.24) is a special case of (3.21). One can also obtain a corollary immediately.

Corollary 3.5. * Let with . If (3.24) has a solution satisfying
**
then
*

#### Acknowledgments

This work is partially supported by the NNSF (no. 11171351) of China and by Scientific Research Fund of Hunan Provincial Education Department (no. 10C0655 and no. 11A095).

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