# Gronwall-Bellman-Type Integral Inequalities and Applications to BVPs

- Chur-Jen Chen
^{1}, - Wing-Sum Cheung
^{1}Email author and - Dandan Zhao
^{1}

**2009**:258569

**DOI: **10.1155/2009/258569

© Chur-Jen Chen et al. 2009

**Received: **9 May 2008

**Accepted: **29 January 2009

**Published: **5 February 2009

## Abstract

We establish some new nonlinear Gronwall-Bellman-Ou-Iang type integral inequalities with two variables. These inequalities generalize former results and can be used as handy tools to study the qualitative as well as the quantitative properties of solutions of differential equations. Example of applying these inequalities to derive the properties of BVPs is also given.

## 1. Introduction

Inequality (1.2) provides an explicit bound to the unknown function and hence furnishes a handy tool in the study of quantitative and qualitative properties of solutions of differential and integral equations. Because of its fundamental importance over the years many generalizations and analogous results of (1.2) have been established (see, e.g., [1–20]). Among various of Gronwall-Bellman-type inequalities, a very useful one is the following.

Theorem 1.1 (see [21]).

for all .

Inequality (1.4) is called Ou-Iang's inequality, which was established by Ou-Iang during his study of the boundedness of certain kinds of second-order differential equations.

Recently, Pachpatte established the following generalization of Ou-Iang-type inequality.

Theorem 1.2 (see [20]).

for all , where is the inverse function of and is chosen such that for all .

Bainov-Simeonov and Lipovan gave the following Gronwall-Bellman-type inequalities, which are useful in the study of global existence of solutions of certain integral equations and functional differential equations.

Theorem 1.3 (see [1]).

for all , where is defined as in Theorem 1.2 and is chosen such that for all .

Theorem 1.4 (see [13]).

for all , where is defined as in Theorem 1.2, and is chosen such that for all .

Very recently, the above results have been further generalized by Cheung to the following.

Theorem 1.5 (see [7]).

Let and be constants. Let , , , and be functions satisfying

(i) are nondecreasing and ;

(ii) is nondecreasing with for .

In this paper, we intend to establish some new nonlinear Gronwall-Bellman-Ou-Iang type integral inequalities with two variables. The setup is basically along the line of [7] but this is by all means a nontrivial improvement of the results there. Examples are also given to illustrate the usefulness of these inequalities in the study of qualitative as well as the quantitative properties of solutions of BVPs.

## 2. Main Results

Throughout this paper, are two fixed numbers. Let , , and , here we allow or to be . We denote by the set of all -times continuously differentiable functions of into , and . Partial derivatives of a function are denoted by , and so forth. The identity function will be denoted as and so, in particular, is the identity function of onto itself.

Note that we allow and to be here.

Theorem 2.1.

Suppose . Let be a constant. If , , , and are functions satisfying

(i) are nondecreasing and ;

(ii) is nondecreasing with for ;

(iii) is strictly increasing with and as ;

Proof.

for all

Corollary 2.2.

Suppose . Let be a constant. If , , , and are functions satisfying

(i) are nondecreasing and ;

(ii)h is strictly increasing with and as ;

Proof.

Let , then . The corollary now follows immediately from Theorem 2.1.

Remark 2.3.

It is easily seen that Theorem 2.1 generalizes Theorems 1.3 and 1.4.

Theorem 2.4.

Suppose . Let be a constant. If , , , and are functions satisfying

(i) are nondecreasing and ;

(ii) is nondecreasing with for ;

(iii) and are strictly increasing with as ;

(iv)for any ,

Proof.

for all

Since is arbitrary, this concludes the proof of the theorem.

Corollary 2.5.

Suppose . Let be a constant. If , , and are functions satisfying

(i) are nondecreasing and ;

(ii) is nondecreasing with for ;

(iii) and are strictly increasing with as ;

(iv)for any ,

Remark 2.6.

If we choose then Corollary 2.5 reduces to Theorem 1.2.

Theorem 2.7.

Suppose . Let be a constant. If , , , and are functions satisfying

(i) are nondecreasing and ;

(ii) is nondecreasing with for ;

(iii) , , and , are strictly increasing with when and as ;

(iv)for any ,

for all , where are the same as in Theorem 2.4, , and is chosen such that for all

Proof.

that is,

for all .

Remark 2.8.

Theorem 2.7 is a generalization of Theorem 1.5.

## 3. Application to Boundary Value Problems

where is defined as in Theorem 2.1, , , and are given.

Our first result deals with the boundedness of solutions.

Theorem 3.1.

Consider BVP (3.1). If

(i) for some ;

(ii) for some ,

then all positive solutions to BVP (3.1) satisfy

In particular, if is bounded on , then every solution to BVP (3.1) is bounded on .

Proof.

The next result is about the quantitative property of solutions.

Theorem 3.2.

for some , then (BVP) (3.1) has at most one solution on .

Proof.

Finally, we investigate the continuous dependence of the solutions of BVP (3.1) on the functional and the boundary data and . For this, we consider the following variation of BVP (3.1):

where is defined as in Theorem 2.1, , , and are given.

Theorem 3.3.

Consider BVP (3.1) and BVP (3.13). If

(i) for some ;

(ii) ;

where is as defined in Theorem 3.1. Hence depends continuously on , , and .

Proof.

Let and be solutions to BVP (3.1) and BVP (3.13), respectively. Then

for some for all . Therefore, depends continuously on , , and .

Remark 3.4.

thus , and so Theorem 3.2 can be viewed as a corollary of Theorem 3.3.

## Declarations

### Acknowledgments

The authors express their gratitude to the referees for their careful reading and many useful comments and suggestions. Research is supported in part by the Research Grants Council of the Hong Kong SAR, China (Project no. HKU7016/07P).

## Authors’ Affiliations

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