- Research Article
- Open Access

# Some Nonlinear Weakly Singular Integral Inequalities with Two Variables and Applications

- Hong Wang
^{1}and - Kelong Zheng
^{1}Email author

**2010**:345701

https://doi.org/10.1155/2010/345701

© Hong Wang and Kelong Zheng. 2010

**Received:**23 October 2010**Accepted:**22 December 2010**Published:**29 December 2010

## Abstract

Some nonlinear weakly singular integral inequalities in two variables which generalize some known results are discussed. The results can be used as powerful tools in the analysis of certain classes of differential equations, integral equations, and evolution equations. An example is presented to show boundedness of solution of a differential equation here.

## Keywords

- Differential Equation
- Integral Equation
- Linear Case
- Type Inequality
- Functional Differential Equation

## 1. Introduction

and the estimates of solutions are given, respectively. In [4], Medveď also generalized his results to an analogue of the Wendroff inequalities for functions in two variables. From then on, more attention has been paid to such inequalities with singular kernel (see [5–9]). In particular, Ma and Yang [8] used a modification of Medveď method to obtain pointwise explicit bounds on solutions of more general weakly singular integral inequalities of the Volterra type, and later Ma and Pečarić [9] used this method to study nonlinear inequalities of Henry type. Recently, Cheung et al. [10] applied the modified Medveď method to investigate some new weakly singular integral inequalities of Wendroff type and applications to fractional differential and integral equations.

Our results can generalize some known results and be used more effectively to study the qualitative properties of the solutions of certain partial differential and integral equations. Moreover, an example is presented to show the usefulness of our results.

## 2. Main Result

In what follows, denotes the set of real numbers, and . denotes the collection of continuous functions from the set to the set . and denote the first-order partial derivatives of with respect to and , respectively.

Before giving our result, we cite the following definition and lemmas.

Definition 2.1 (see [8]).

Let be an ordered parameter group of nonnegative real numbers. The group is said to belong to the first-class distribution and is denoted by if conditions , , and are satisfied; it is said to belong to the second-class distribution and is denoted by if conditions , and are satisfied.

Lemma 2.2 (see [8]).

where ( ) is well-known -function and .

Lemma 2.3 (see [8]).

Suppose that the positive constants *α*,
,
,
, and
satisfy the following conditions:

(1)if , ;

(2)if , .

are valid.

Assume that

(A_{1})
and
;

(A_{2})
is nondecreasing and
.

Let and .

Theorem 2.4.

Proof.

Next, for convenience, we introduce indices , . Denote that if , then let and ; if , then let and . Then holds for .

and is given in Lemma 2.3 for .

_{2}), is strictly increasing so its inverse is continuous and increasing in its corresponding domain. Replacing and by and , we have

for and .

Finally, considering two situations for
and using parameters *α*, *β*,
to denote
,
,
, and
in the above inequality, we can obtain the estimations, respectively. we omit the details here.

Remark 2.5.

*Medveď*[4, *Theorem* 2.2]*investigated the special case*(
,
)*of inequality ( 1.2 ) under the assumption that "*
*satisfies the condition*
*." However, in our result, the*
*condition is eliminated. If we take*
*and*
*, then we can obtain the result of linear case*[4, *Theorem* 2.4]*.*

Remark 2.6.

*Let*
*, then*
*or*
*. Therefore, if we take*
*, the formula*(2.6)*in*[10]*is the special case of inequality*(1.2),*and we can obtain more concise results than*(2.7)*and*(2.9)*in*[10]*. Moreover, here the condition*
*also can be eliminated.*

Remark 2.7.

*When*
*does not belong to*
*or*
*, there are some technical problems which we do not discuss here.*

## 3. Some Corollaries

Corollary 3.1.

- (1)
for ,

for , , where , , are defined as in Theorem 2.4,

- (2)
for ,

for , , where , , are defined as in Theorem 2.4.

Proof.

Clearly, inequality (3.1) is the special case of (1.2). Taking , we can get (3.1).

Therefore, the positive numbers and in (2.6) and (2.10) can be taken as , and the results can be obtained by simple computation. We omit the details.

Corollary 3.2.

Proof.

(i)For ,

Since inequality (3.19) is similar to (2.12), we can repeat the procedure of proof in Theorem 2.4 and get (3.9).

(ii)As for the case that , the proof is similar to the argument in the proof of case (i) with suitable modification. We omit the details.

Remark 3.3.

*When*
*,*
*or*
*,*
*, we can get the results which are similar to that in Corollary 3.2 and omit them here.*

## 4. Application

In this section, we will apply our result to discuss the boundedness of certain partial integral equation with weakly singular kernel.

which implies that in (4.1) is bounded.

## Declarations

### Acknowledgments

This work is supported by Scientific Research Fund of Sichuan Provincial Education Department (no. 09ZC109). The authors are very grateful to the referees for their helpful comments and valuable suggestions.

## Authors’ Affiliations

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## Copyright

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