Abstract and Applied Analysis

Volume 2013 (2013), Article ID 830595, 8 pages

http://dx.doi.org/10.1155/2013/830595

Research Article

On Inequalities of Lyapunov for Two-Dimensional Nonlinear Dynamic Systems on Time Scales

Qiao-Luan Li,¹ Wing-Sum Cheung,² and Xu-Yang Fu¹

¹College of Mathematics & Information Science, Hebei Normal University, Shijiazhuang 050024, China

²Department of Mathematics, The University of Hong Kong, Hong Kong

Received 8 July 2013; Accepted 9 October 2013

Academic Editor: Allan Peterson

Copyright © 2013 Qiao-Luan Li 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 Lyapunov-type inequalities for two-dimensional nonlinear dynamic systems on time scales. As for application, boundedness of the Emden-Fowler-type equation is proved.

1. Introduction

In this paper, we establish some Lyapunov-type inequalities for the following two-dimensional nonlinear dynamic system:

wherea time scale,denotes the delta derivative with respect to, anddenotes the delta derivative with respect to.

Lyapunov-type inequalities have proven to be very useful in the study of qualitative behavior of solutions such as oscillation, disconjugacy, and eigenvalue problems for differential equations and difference equations. Since the appearance of Lyapunov's fundamental paper [1], considerable attention has been given to various extensions and improvements of the Lyapunov-type inequality from different viewpoints [2–8]. Although Lyapunov-type inequalities are well developed for the continuous cases, their time scale versions are still in early stages and are worth due attention.

Recently, He et al. in [2] considered the linear Hamiltonian system

and obtained several useful Lyapunov-type inequalities.

Chen et al. in [3] considered the nonlinear system

and obtained some interesting Lyapunov-type inequalities for partial differential equations.

In this paper, under the assumption of existence of a nontrivial solutionto the 2-dimensional nonlinear dynamic system (1), some new and interesting Lyapunov-type inequalities are established.

2. Main Results

Throughout this paper, the following mild and natural conditions are assumed:

(i),are real constants,(ii),  ,are rd-continuous functions such thatfor, where, andis the forward jump operator; that is,.

Theorem 1. If the nonlinear dynamic system (1) has a real solutionwhich is not identically zero onsatisfyingandfor all, wherewith,  , then

where,, and.

Proof. From the conditionsandis not identically zero on, there existssuch that.

Multiplying the first equation of (1) byand the second one byand adding up, we get

and, hence,

Integrating the left hand side of (6) overfromtoand then overfromto, we get

Noting that, we have

On the other hand, integrating the first equation of (1) overfromtoand then overfromto, we get

By the boundary conditions on, it is elementary to verify that

and so

Hence,

By similar arguments, we easily get

Summing (12) and (13) and by Hölder's inequality with indicesand, we obtain

In view of (8), we have

and so

The proof is complete.

Remark 2. It is interesting to note that whenTheorem 1 reduces to Theoremof [3].

Theorem 3. If the nonlinear dynamic system (1) has a real solutionwhich is not identically zero onsatisfying,,, andfor all, wherewith,, then

whereand,are as defined in Theorem 1.

Proof. Choosesuch that. Note that. From (6) and

we have

By the boundary conditions on, it is elementary to check that

So

For the fixed, by, there existssuch that

Integrating the first equation of (1) overfromto, we obtain

Multiplying (23) byand noting that, we get

Letting, we get

By (22) and (25), we obtain

Substituting (26) into (21), we get

Immitating the arguments from (21) to (27) step by step, we have

By, there existssuch that

Integrating the first equation of (1) overfromtoand using, we get

and, hence,

Let

Then, (27) can be written as

and (31) can be written as

It follows from (33) and (34) that

Using (19),,, and Hölder's inequality, we have

Therefore,

The proof is complete.

Theorem 4. Suppose thatandare conjugate exponents; that is,. If the nonlinear dynamic system (1) has a real solutionwhich is not identically zero onsuch thatandfor all, wherewith, then

Proof. By (6) and the conditionsfor all, we have

So

Fixsuch that. Integrating the first equation of (1) overfromtoand then overfromto, we get

and so by Hölder's inequality with indicesandwe have

Hence,

The proof is complete.

Theorem 5. Suppose thatandare conjugate exponents. If the nonlinear dynamic system (1) has a real solutionwhich is not identically zero onsuch thatandfor all, wherewith, then

Proof. By (6) and the assumption thatfor all, we get

So

Fixsuch that. Integrating the first equation of (1) overfromtoand then overfromto, we get

and so by Hölder's inequality with indicesandwe have

Hence,

The proof is complete.

Remark 6. Analogously, we can also consider the cases (i)and (ii). Similar results to those in Theorem 4 and Theorem 5 can easily be arrived at. The detailed proofs are omitted here.

Next, we exhibit an application of our results. Consider the following special case of (1):

Definition 7. A nontrivial solutionof the dynamic equation (50) defined onis said to be proper if

for all,  A proper solutionof the dynamic system (50) is called weakly oscillatory if at least one argument has a sequence of zeros diverging to.

Theorem 8. Assume thatis bounded onfor a fixed,is bounded onfor a fixed,

then every weakly oscillatory proper solution of (50) is bounded on

Proof. Letbe a nontrivial weakly oscillatory proper solution of (50) onandhave a sequence of zeros diverging to. Suppose that; then, for any positive constant, there existssuch that,Sinceis an oscillatory solution, there existssuch that,on, and. From (52), we can choosesufficiently large such that

By Theorem 1, we have

and so

which contradicts. Hence, there exists a positive constantsuch that.

Integrating the second equation of the dynamic equation (50) overfromtoand then overfromto, respectively, we get

Sinceis bounded onfor a fixedandis bounded onfor a fixed, we get

Since,is bounded. Sois bounded onThe proof is complete.

Acknowledgments

The first author's research was supported by NNSF of China (11071054) and Natural Science Foundation of Hebei Province (A2011205012). The corresponding author's research was partially supported by an HKU URG Grant.

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