# Nonlinear Retarded Integral Inequalities with Two Variables and Applications

- Wu-Sheng Wang
^{1, 2}Email author, - Zizun Li
^{1}, - Yong Li
^{3}and - Yong Huang
^{3}

**2010**:240790

https://doi.org/10.1155/2010/240790

© Wu-Sheng Wang et al. 2010

**Received: **8 June 2010

**Accepted: **6 October 2010

**Published: **11 October 2010

## Abstract

We consider some new nonlinear retarded integral inequalities with two variables, which extend the results in the work of W.-S. Wang (2007), and the one in the work of Y.-H. kim (2009). These inequalities include not only a nonconstant term outside the integrals but also more than one distinct nonlinear integrals without assumption of monotonicity. Finally, we give some applications to the boundary value problem of a partial differential equation for boundedness and uniqueness.

## Keywords

## 1. Introduction

for all , where is a constant.

for all , where is a constant.

The purpose of the present paper is to establish some new nonlinear retarded integral inequalities of Gronwall-Bellman type with two variables. We can demonstrate that inequalities (1.4), (1.5), and (1.7), considered in [9, 12, 16], respectively, can also be solved with our results. We also apply our results to study the boundedness and uniqueness of the solutions of the boundary value problem of a partial differential equation.

## 2. Main Result

Throughout this paper, denotes the set of real numbers, and are given numbers. , , are the subsets of and . For any , let denote the subset of . denotes the set of continuous differentiable functions of into .

Our inequality (2.1) not only includes a nonconstant term outside the integrals but also more than one distinct nonlinear integral without assumption of monotonicity. When , and , our inequality (2.1) reduces to (1.7) studied in [12]. When , and our inequality (2.1) reduces to (1.5) studied in [16].

Suppose that

is a strictly increasing continuous function on , ;

all are continuous functions on and positive on ;

on , and is nondecreasing in each variable;

and are nondecreasing such that and on , and on ;

all are nonnegative functions on .

Obviously, both and are strictly increasing and continuous functions. Letting denote inverse function, respectively, then both and are also continuous and increasing functions.

Then and are nonnegative and nondecreasing in for each fixed and satisfy , , .

Theorem 2.1.

Corollary 2.2.

Corollary 2.3.

Theorem 2.4.

Corollary 2.5.

## 3. Proofs and Remarks

Proof of Theorem 2.1.

for all . This proves that (3.10) is true for .

for all The claim in (3.10) is proved by induction.

for all . Hence, we obtain the estimation of the unknown function in the auxiliary inequality (3.2).

for all , . Since and and are arbitrarily chosen, this proves (2.7).

for all . Letting , we obtain (2.7) because of continuity of in and continuity of , and for . This completes the proof.

The proofs of Corollary 2.2, 2.3, 2.5 and Theorem 2.4 are similar to the argument in the proofs of Theorem 2.1 with appropriate modification. We omit the details here.

Remark 3.1.

When and , Corollary 2.2 reduces to Theorem in [9].

Remark 3.2.

When and , Corollary 2.3 reduces to Theorem of Wang [16].

Remark 3.3.

When , Theorem 2.4 reduces to Theorem of Kim [12].

## 4. Applications

for all , where is defined as in Section 2, , are defined in , is a continuous and strictly increasing odd function on , satisfying , for , , , , and are nondecreasing continuous functions, and the ratio is also nondecreasing, and for .

In the following corollary, we firstly apply our result to discuss boundedness on the solution of problem (4.1).

Corollary 4.1.

Proof.

we obtain the estimation of as given in (4.5).

for all . Then every solution of (4.1) is bounded on .

Next, we discuss the uniqueness of the solutions of (4.1).

Corollary 4.2.

for and , where is defined as in Section 2, is a constant, , , are continuous nondecreasing with the nondecreasing ratio such that for all , and , , and is a strictly increasing odd function satisfying for all . Then, (4.1) has at most one solution on

Proof.

and denotes the inverse function of .

Thus, from (4.5), we deduce that , implying that , for all , since is strictly increasing. The uniqueness is proved.

## Declarations

### Acknowledgments

The authors are very grateful to the editor and the referees for their helpful comments and valuable suggestions. This work is supported by the Natural Science Foundation of Guangxi Autonomous Region (0991265), the Scientfic Research Foundation of the Education Department of Guangxi Autonomous Region (200707MS112), the Key Discipline of Applied Mathematics (200725) and the Key Project (2009YAZ-N001) of Hechi University of China.

## Authors’ Affiliations

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