- Research Article
- Open Access

# On Lyapunov-Type Inequalities for Two-Dimensional Nonlinear Partial Systems

- Lian-Ying Chen
^{1}, - Chang-Jian Zhao
^{1}Email author and - Wing-Sum Cheung
^{2}

**2010**:504982

https://doi.org/10.1155/2010/504982

© Lian-Ying Chen et al. 2010

**Received:**27 February 2010**Accepted:**23 June 2010**Published:**11 July 2010

## Abstract

We establish a new Laypunov-type inequality for two nonlinear systems of partial differential equations and the discrete analogue is also established. As application, boundness of the two-dimensional Emden-Fowler-type equation is proved.

## Keywords

- Partial Differential Equation
- Nonlinear System
- Triangle Inequality
- Nontrivial Solution
- Real Solution

## 1. Introduction

In this paper, we obtain new Lyapunov-type inequalities for the two-dimensional nonlinear system and discrete nonlinear system, respectively.

## 2. The Lyapunov-Type Integral Inequality for the Two-Dimensional Nonlinear System

We shall assume the existence of nontrivial solution of the system (2.1), and furthermore, (2.1) satisfies the following assumptions (i), (ii), and (iii):

(i) , are real constants;

(ii) , are continuous functions such that for ;

(iii) is a continuous function.

Theorem 2.1.

where , , and is the nonnegative part of

Proof.

where .

The proof is complete.

Remark 2.2.

This is just a new Lyapunov-type inequality which was given by Tiryaki et al. [8].

## 3. The Lyapunov-Type Discrete Inequality for the Two-Dimensional Nonlinear System

where denotes the forward difference operator for that is, and denotes the forward difference operator for that is, We shall assume the existence of nontrivial solution of the system (3.1), and furthermore, (3.1) satisfies the following assumptions ( ), ( ), and ( ):

(i) are real constants;

(ii) , are real-valued functions such that for all ;

(iii) is a real-valued function for all

Theorem 3.1.

where , and

Proof.

This completes the proof.

Remark 3.2.

This is just a new Lyapunov-type inequality which was given by Ünal et al. [2].

## 4. An application

where is a constant, and are real functions, and for all .

where and .

for any
.A proper solution
of system (4.2) is called *weakly oscillatory* if and only if at least one component has a sequence of zeros tending to
.

Theorem 4.1.

then every weakly oscillatory proper solution of (4.2) is bounded on .

Proof.

where and .

This contradiction shows that is bounded on . Therefore, there exists a positive constant such that for all .

where is a constant.

that is bounden on .

This completes the proof.

## Declarations

### Acknowledgments

This research is supported by National Natural Sciences Foundation of China (10971205). It is also partially supported by the Research Grants Council of the Hong Kong SAR, China (Project no. HKU7016/07P) and an HKU Seed Grant for Basic Research.

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. 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.