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

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.

1. Introduction

In a celebrated paper of 1893, Liapunov [1] proved the following well-known inequality: if is a nontrivial solution of

(1.1)

on an interval containing the points and b such that then

(1.2)

Since the appearance of Liapunov's fundamental paper [1], considerable attention has been given to various extensions and improvements of the Lyapunov-type inequality from different viewpoints [2–7]. In particular, the Lyapunov-type inequalities for the following nonlinear system of differential equations were given in [8]

(1.3)

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

(2.1)

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.

Let the hypotheses hold. If the nonlinear system (2.1) has a real solution such that for and and is not identically zero on , where with , then

(2.2)

where , , and is the nonnegative part of

Proof.

Since and is not identically zero on , we can choose such that . Let . Integrating the first equation of system (2.1) over from to and over from to , respectively, we obtain

(2.3)

On the other hand, we have

(2.4)

Hence,

(2.5)

and similarly, we have

(2.6)

Employing the triangle inequality gives

(2.7)

(2.8)

Summing (2.7) and (2.8), we obtain

(2.9)

By using Hölder inequality on the second integral of the right side of (2.9) with indices and , we have

(2.10)

where .

Therefore, we obtain from (2.9)

(2.11)

On the other hand, we have

(2.12)

Multiplying the first equation of (2.1) by and the second one by , adding the result, and noting , we have

(2.13)

Integrating the left side of (2.13) over from to and over from to , respectively, we get

(2.14)

Now integrating both sides of (2.13) over from to and over from to , respectively, and noting we get

(2.15)

Substituting equality (2.15) by (2.11), we have

(2.16)

Noticing that and , we obtain

(2.17)

The proof is complete.

Remark 2.2.

Let change to in (2.2), and with suitable changes, (2.2) changes to the following result:

(2.18)

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

(3.1)

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.

Let the hypotheses hold. Assume and If the nonlinear system (3.1) has a real solution such that for all and and is not identically zero on then

(3.2)

where , and

Proof.

Let be nontrivial real solution of system (3.1) such that and is not identically zero on . Then multiplying the first equation of (3.1) by and the second one by adding the result, and noting and

(3.3)

we have

(3.4)

Summing the left side of (3.4) over from to and over from to , respectively, we have

(3.5)

Summing both sides of (3.4) over from to and over from to , respectively, and noting we obtain

(3.6)

Noticing that and we have

(3.7)

Choose such that Hence Summing the first equation of (3.1) over from to and over from to , respectively, we obtain

(3.8)

Considering the left side of (3.8) and noting for all we have

(3.9)

Hence,

(3.10)

and similarly, we have

(3.11)

Employing the triangle inequality gives

(3.12)

(3.13)

Summing (3.12) and (3.13), we obtain

(3.14)

On the other hand, using Hölder inequality on the second sum of the right side of (3.14) with indices and we have

(3.15)

where Therefore, from (3.7) and we obtain

(3.16)

Substituting (3.16) to (3.14), we have

(3.17)

Noticing that we get

(3.18)

This completes the proof.

Remark 3.2.

Let change to in (3.2) and with suitable changes, (3.2) changes to the following result:

(3.19)

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

4. An application

Two-dimensional Emden-Fowler-type equation

(4.1)

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

Consider the following special case of system (2.1), which is an equivalent system for the two-dimensional Emden-Fowler-type equation (4.1)

(4.2)

where and .

Obviously Theorem 2.1 for the two-dimensional nonlinear system (2.1) with is satisfied for system (4.2). Therefore, we have

(4.3)

A nontrivial solution of system (4.2) defined on is said to be proper if and only if

(4.4)

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.

If , where , and , is bounded on and is bounded on ,

(4.5)

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

Proof.

Let be any nontrivial weakly oscillatory proper solution of nonlinear system (4.2) on such that has a sequence of zeros tending to . Suppose to the contrary that ; then given any positive number , we can find positive numbers and such that for all . Since is an oscillatory solution, there exist with such that and on . Choose in such that ; in view of (4.5), we can choose and large enough such that for every , ,

(4.6)

Taking th power of both sides of (4.3) and combining (4.6), we obtain

(4.7)

where and .

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

On the other hand, integrating the second equation of system (4.2) over from to and over from to , respectively, we obtain

(4.8)

Notice that is bounded on , is bounded on , and in view of triangle inequality, we have

(4.9)

where is a constant.

Equation (4.9) implies that is bounded on since . It follows from

(4.10)

that is bounden on .

This completes the proof.