Abstract and Applied Analysis

Volume 2014 (2014), Article ID 539701, 9 pages

http://dx.doi.org/10.1155/2014/539701

Research Article

New Explicit Bounds on Gamidov Type Integral Inequalities for Functions in Two Variables and Their Applications

Kelong Cheng¹ and Chunxiang Guo²

¹School of Science, Southwest University of Science and Technology, Mianyang 621010, China

²School of Business, Sichuan University, Chengdu 610064, China

Received 14 February 2014; Accepted 10 June 2014; Published 22 June 2014

Academic Editor: Praveen Agarwal

Copyright © 2014 Kelong Cheng and Chunxiang Guo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Some linear and nonlinear Gamidov type integral inequalities in two variables are established, which can give the explicit bounds on the solutions to a class of Volterra-Fredholm integral equations. Some examples of application are presented to show boundedness and uniqueness of solutions of a Volterra-Fredholm type integral equation.

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 differential and integral equations. During the past few years, an enormous amount of effort has been devoted to the discovery of new types of inequalities and their applications in various branches of ordinary and partial differential and integral equations; see [1–14] and the references given therein.

In particular, Gamidov [6], while studying the boundary value problem for higher order differential equations, initiated the study of obtaining explicit upper bounds on the integral inequalities of the forms

for , under some suitable conditions on the functions involved in (1). Later, such inequalities (usually called Gamidov type inequalities) are also discussed by Bainov and Simeonov [15] and Pachpatte [16, 17]. In [16], Pachpatte established more general Gamidov inequalities as follows:

and also generalized it to a general version in two independent variables [18] as follows:

Based on the results of inequalities (3), the retarded cases with power nonlinearity of inequalities (3) are studied by Ma and Pečarić [19]. As a matter of fact, the results of inequality (2) are not right. In the proof of Theorem 1 in [16] about inequalities (2), by definition and inequalities , they obtained (for details, see [16, page 355]). However, we can see easily that is a function in , while it was treated as a constant in [16].

In this paper, we discuss some linear and nonlinear Gamidov type integral inequalities and obtain new explicit bounds on these inequalities. Moreover, we also give some examples to show boundedness and uniqueness of solutions of a Volterra-Fredholm type integral equation.

2. Linear Gamidov Type Integral Inequalities

In what follows, denotes the set of real numbers, , , , and are given subsets of . Let . denotes the collection of continuous functions from to .

Lemma 1. Assume , , , and

for . If , then the following explicit estimate

holds for .

Proof. Obviously, is a constant. Letting , from (4), we have

Since is positive, then

Integrating for both sides of (7) on , we have

Consequently, it follows from the inequality above that

where . Substituting the inequality above into (6), we get the explicit estimate (5) for .

Theorem 2. Assume , , , , , and , , are nondecreasing in and . If satisfies

then, under the condition that

the following explicit estimate

holds for .

Proof. Fixing any arbitrary , then for , from (10), we have

where we apply that , , and are nondecreasing in and .

Define a function , by the right side of (13). Then, is positive and nondecreasing in and , and

Furthermore, we have

Since is nondecreasing in , from (16), one gets

Keeping fixed in (17), setting , and integrating it from to , we get

It follows from (14) and (15) that

where

Now, a suitable application of Lemma 1 yields

Since the inequality above holds for all , taking and in (21), we have

for , where

Substituting (23) into (22) and simplifying it, we get

Considering that and are arbitrary, we replace and by and , respectively, and get the desired estimate (12). The proof is complete.

Remark 3. If , , and are not nondecreasing, our result also holds, since we can replace it by , , and .

Remark 4. As for inequality (3) discussed by Pachpatte [18] or a more general case as follows:

there are some technical problems which we do not discuss here. Clearly, inequalities (4) are the special case of inequalities (25); that is, and .

3. Nonlinear Gamidov Type Integral Inequalities

In this section, we discuss some integral inequalities with some power nonlinear terms.

Lemma 5 (see [20]). Let , , and . Then

for any .

Theorem 6. Assume that , , , , and are defined as in Theorem 2. If satisfies

where , , and , , and are constants, then, under the condition that

the following explicit estimate

holds for , where

for any .

Proof. Define a function by

for . Then, from (27), we have

or

Applying Lemma 5, for any , we get

Substituting (34) into (31), we get

which is similar to inequality (10), where , , , , and are defined as in (30). It is easy to see that , , and are nonnegative, continuous, and nondecreasing for .

A suitable application of Theorem 2 to (35) yields

Substituting (36) into (33), we can get (29).

Remark 7. In [19], Ma and Pečarić discussed the retarded Gamidov type integral inequality:

As mentioned above, the results of inequality (3) obtained by Pachpatte [18] are not right. So the results in [19] are not valid.

When ,   in Theorem 6, an Ou-Iang version of Gamidov type inequality is obtained as follows.

Corollary 8. Assume that , , , , and are defined as in Theorem 2. If satisfies

Then, under the condition that

the following explicit estimate

holds for , where

for any .

When , in Theorem 6, we can get an interesting result as follows.

Corollary 9. Assume that , , , , and are defined as in Theorem 2. If satisfies

then, under the condition that

the following explicit estimate

holds for , where

for any .

4. Applications

In this section, we present some applications of our results to study boundedness and uniqueness of solutions of a Volterra-Fredholm integral equation in two independent variables.

Considering the following integral equation:

for , where , , , , , , and is a constant.

Theorem 10. Assume that the functions and in (46) satisfy the conditions

where and are defined as in Theorem 2, and satisfying , are constants. If

and is a solution of (46) and (47), then

for and any , where , , , , and are defined as in Theorem 6.

Proof. Letting be a solution of (46), from (47), we have

An application of the inequality given in Theorem 6 to (50) yields (49). The right-hand side of (49) gives us the bound on the solution of (46) in terms of the unknown functions.

Considering the following integral equation:

for , where , , , , , .

Theorem 11. Assume that the functions and in (46) satisfy the conditions

where and are defined as in Theorem 2, is a constant. If

where and are defined as in Corollary 9 and is a solution of (51) and (52), then (51) has at most one solution.

Proof. Let and be two solutions of (51); that is,

From (51) and (54), we have

According to Corollary 9, we obtain that , which implies for .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are very grateful to the referees for their helpful comments and valuable suggestions. This work is supported by the Doctoral Program Research Funds of Southwest University of Science and Technology (no. 11zx7129) and the Fundamental Research Funds for the Central Universities (no. skqy201324).

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