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# Nonlinear Retarded Integral Inequalities with Two Variables and Applications

*Journal of Inequalities and Applications*
**volumeÂ 2010**, ArticleÂ number:Â 240790 (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.

## 1. Introduction

Integral inequalities that give explicit bounds on unknown functions provide a very useful and important device in the study of many qualitative as well as quantitative properties of solutions of partial differential equations, integral equations, and integrodifferential equation. One of the best known and widely used inequalities in the study of nonlinear differential equations is Gronwall inequality [1], which states that if and are nonnegative continuous functions on the interval satisfying

where is a nonnegative constant, then we have

Since the inequality (1.2) provides an explicit bound of the unknown function it furnishes a handy tool in the study of various properties of solutions of differential equations. Because of its fundamental importance, several generalizations and analogous results of Gronwall inequality [1, 2] and its applications have attracted great interests of many mathematicians (e.g., [3â€“5]). Some recent works can be found, for example, in [6â€“17] and some references therein. In 2005, Agarwal et al. [6] investigated the inequality

In 2006, Cheung [9] studied the inequality

for all , where is a constant.

In 2007, Wang [16] discussed the retarded integral inequality

for all .

In 2008, Agarwal et al. [7] discussed the retarded integral inequality

for all , where is a constant.

In 2009, Kim [12] obtained the explicit bound of the unknown function of the following inequality:

for all .

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 .

Consider the following inequality:

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 ;

is a constant;

all are nonnegative functions on .

Firstly, we technically consider a sequence of functions , which can be calculated recursively by

Moreover, we define the following functions:

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

Let

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

Theorem 2.1.

Suppose that hold and is a nonnegative function on satisfying (2.1). Then

for , where

is arbitrarily given on the boundary of the planar region

Corollary 2.2.

Let and be as defined in Theorem 2.1. Suppose that are constants. If

for all , then

for all , where and are defined by (2.5) and (2.6), and

denotes the inverse function of , and lies on the boundary of the planar region

Corollary 2.3.

Let and be as defined in Theorem 2.1. Supposing that

for all , then

for all , where

is defined by (2.5), , are as defined in (2.3) and (2.4), respectively, denote the inverse functions of and . lies on the boundary of the planar region

Theorem 2.4.

Suppose that hold and is a nonnegative function on satisfying

Then

for , where

for , and is arbitrarily given on the boundary of the planar region

Corollary 2.5.

Suppose that hold and is a nonnegative function on satisfying

where are nonnegative functions on . Then

for , where

for , and is arbitrarily given on the boundary of the planar region

## 3. Proofs and Remarks

Proof of Theorem 2.1.

Obviously, the sequence defined by in (2.2) is nondecreasing nonnegative functions and satisfies , . Moreover, the ratios , are all nondecreasing. From (2.1), (2.2) and (2.5), (2.6), we have

We first discuss the case that for all . Consider the auxiliary inequality

for all , where and are chosen arbitrarily. Let denote the function on the right-hand side of (3.2), which is a nonnegative and nondecreasing function on and . Then, we get the equivalent form of (3.2)

Since is nondecreasing and satisfies for . By the definition of , hypothesis , the monotonicity of and , and (3.3), we have

From (3.4), we have

Keeping fixed in (3.5), setting , integrating both sides of (3.5) with respect to from to , and using the definition of in (2.3), we have

for all , where

Let

From (3.6), we have

for all . We claim that the unknown function in (3.9) satisfies

for all , where

Now, we prove (3.10) by induction. For , let denote the function on the right-hand side of (3.9), which is a nonnegative and nondecreasing function on , and . Then we have

for all . From (3.13), we have

Keeping fixed in (3.14), setting , integrating both sides of (3.14) with respect to from to , and using the definition of in (2.4), we have

for all . Using , from (3.15), we obtain

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

Next, we make the inductive assumption that (3.10) is true for . Now, we consider

for all . Let denote the nonnegative and nondecreasing function on the right-hand side of (3.17). Then and

Let

By (2.2), we see that each , , is a nondecreasing function. Then, we have

for all . Keeping fixed in (3.20), setting , integrating both sides of (3.20) with respect to from to , and using the definition of in (2.4), we have

for all . Let

Using (3.22) and (3.23), from (3.21), we have

It has the same form as (3.9). Let . Since , and are continuous, nondecreasing, and positive on , each is continuous, nondecreasing, and positive on . Moreover,

which are also continuous, nondecreasing, and positive on . Therefore, the inductive assumption for (3.9) can be used to (3.24), and then we have

for all , where

We note that

Thus, from (3.18), (3.22), (3.26), and (3.29), we have

for all . We can prove that the term of in (3.30) is just the same as defined in (3.12). Let . By (3.23), we have

Then using (3.28) and (3.29), we get

This proves that in (3.30) is just the same as defined in (3.12). Therefore, from (3.29), (3.30), and (3.32), we obtain

. The relations of (3.33) imply that in (3.26) and (3.28)

. Hence, (3.30) can be equivalently written as

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

Therefore, by (3.3), (3.8), and (3.10), we have

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

Letting , , from (3.36), we have

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

The remainder case is that for some . Let

where is an arbitrary small number. Obviously, , for all . Using the same arguments as above, where is replaced with , we get

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

In this section, we apply our results to study the boundedness and uniqueness of the solutions of boundary value problem to a partial differential equation. We consider the partial differential equation with the initial boundary conditions:

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 â€‰â€‰.

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

Corollary 4.1.

Assume that is a continuous function for which there exist a constant , nonnegative functions , , such that

for all , and is nondecreasing in each variable. If is any solution of problem (4.1) with condition (4.2), then

for all , where

and are as defined in Theorem 2.1. lies on the boundary of the planar region

Proof.

It is easy to see that the solution of (4.1) satisfies the following equivalent integral equation:

By(4.3),(4.4), and (4.8), we have

Since , (4.9) is the form of (2.14). Applying Corollary 2.3 to inequality (4.9), using the relation

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

Corollary 4.1 gives a condition of boundedness for solutions, concretely. If there is a ,

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.

Additionally, assume that

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.

Let and be two solutions of (4.1). By (4.8) and (4.12), we have

for all , which is an inequality of the form (2.14), where is an arbitrary small number. Applying Corollary 2.2, we obtain an estimation of the difference in the form (4.5). Namely,

for all , where

and denotes the inverse function of .

Furthermore, by the definition of , we conclude that

Letting , it follows that

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

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## 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.

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Wang, WS., Li, Z., Li, Y. *et al.* Nonlinear Retarded Integral Inequalities with Two Variables and Applications.
*J Inequal Appl* **2010**, 240790 (2010). https://doi.org/10.1155/2010/240790

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