Journal of Applied Mathematics

Volume 2014 (2014), Article ID 865136, 13 pages

http://dx.doi.org/10.1155/2014/865136

Research Article

A Generalized Nonlinear Volterra-Fredholm Type Integral Inequality and Its Application

Limian Zhao,¹ Shanhe Wu,² and Wu-Sheng Wang¹

¹School of Mathematics and Statistics, Hechi University, Yizhou, Guangxi 546300, China

²Department of Mathematics and Computer Science, Longyan University, Longyan, Fujian 364012, China

Received 13 March 2014; Accepted 16 April 2014; Published 21 May 2014

Academic Editor: Kuppalapalle Vajravelu

Copyright © 2014 Limian Zhao et al. 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.

We establish a new nonlinear retarded Volterra-Fredholm type integral inequality. The upper bounds of the embedded unknown functions are estimated explicitly by using the theory of inequality and analytic techniques. Moreover, an application of our result to the retarded Volterra-Fredholm integral equations for estimation is given.

1. Introduction

Gronwall-Bellman inequality [1, 2] is an important tool in the study of existence, uniqueness, boundedness, oscillation, stability, invariant manifolds, and other qualitative properties of solutions of differential equations and integral equation. A lot of its generalizations in various cases can be found from the literature (e.g., [3–7]). During the past few years, some investigators have established a lot of useful and interesting integral inequalities in order to achieve various goals; see [8–18] and the references cited therein.

Gronwall-Bellman inequality [1, 2] can be stated as follows. If and are nonnegative continuous functions on an interval satisfying

for some constant , then

In 2004, Pachpatte [9] has discussed the linear Volterra-Fredholm type integral inequality with retardation:

In 2011, Abdeldaim and yakout [17] studied a new integral inequality of Gronwall-Bellman-Pachpatte type:

In this paper, on the basis of [9, 17], we discuss a new retarded nonlinear Volterra-Fredholm type integral inequality:

where is a constant. The upper bound estimation of the unknown function is given by integral inequality technique, such as change of variable, amplification method, differential and integration, inverse function, and the dialectical relationship between constants and variables. Furthermore, we apply our result to retarded nonlinear Volterra-Fredholm type equations for estimation.

2. Main Result

Throughout this paper, denotes the set of real numbers, , , denotes the class of continuously differentiable functions defined on set with range in the set , denotes the class of continuous functions defined on set with range in the set , and denotes the derived function of a function .

We give the following notations used to simplify the details of presentation.

We technically define a sequence of functions by in (5), which can be defined recursively by

Obviously, for all , the function is increasing and the sequence consists of nondecreasing nonnegative functions and satisfies , . Moreover,

as defined in [4] for comparison of monotonicity of functions, because the ratios , , are all nondecreasing.

For given constant , we define functions

which are strictly increasing. When there is no confusion, we simply let denote and denote its inverse.

We define functions :

We define function

Theorem 1. Suppose that is nondecreasing with and on ; all are continuous functions with for , , . Suppose that the function is increasing and has a solution for . If satisfies (5), then

where are inverse functions of, respectively.

Proof. From (5) and (6), we have

for all . Let denote the function on the right-hand side of (13), which is a positive and nondecreasing function on . Then (13) is equivalent to

Differentiating with respect to , using (14), we have

by the monotonicity of and and the property of . From (16), we have

Integrating both sides of the above inequality from to , we obtain

for ; is chosen arbitrarily, where is defined by (8).

Let denote the function on the right-hand side of (18), which is a positive and nondecreasing function on . Then (18) is equivalent to

Differentiating with respect to , using (19), we have

by the monotonicity of and the property of . From (21), we have

for all . From (22), we have

for all , where is defined by (9). Repeating the same derivation as in (19), (23), and so on, we obtain

for all , where is defined by (9).

Let denote the function on the right-hand side of (24), which is a positive and nondecreasing function on . Then (24) is equivalent to

Differentiating with respect to , we have

for all . From (27), using (25), we have

for all , by the monotonicity of , and and the property of . Integrating both sides of the above inequality from to , we obtain

for all , where is defined by (9). Let denote the function on the right-hand side of (29), which is a positive and nondecreasing function on . Then (29) is equivalent to

Differentiating with respect to , using (30), we have

for all . From (32), we have

for all . Integrating both sides of the above inequality from to , we obtain

for all . From (19), (25), (30), and (34), we have

for all . Substituting (20), (26), and (31) into (35), we have

Since is chosen arbitrarily, we have

By the definition of and (15), we have

From (37) and (38), we have

or

By the definition of , the assumption of Theorem 1, and (40), we observe that

Since is increasing, from the last inequality and (14), we have the desired estimation (12).

We define the following functions:

for all , where , are defined by (8) and (9), respectively.

Corollary 2. Let , , , , , , be as in Theorem 1. Suppose that the function is increasing and has a solution for . If satisfies (5), then

where are inverse functions of, respectively.

3. Application

In this section, we apply our result in Theorem 1 to investigate the retarded Volterra-Fredholm integral equations:

for , where , is nondecreasing with ,,,,. Let ; then ,. Since , is an increasing and invertible function.

The following theorem gives the bound on the solution of (44).

Theorem 3. Suppose that , in (44) satisfy the conditions

where ,,, and are as in Theorem 1; let . Assume that the function

is increasing and has a solution for . If is a solution of (44), then

where ,,, and are as in Theorem 1.

Proof. Using the condition (45), we have

for , where several changes of variables are made. Applying the result of Theorem 1 to the last inequality, we obtain the desired estimation (47).

Conflict of Interests

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

Acknowledgments

The present investigation was supported, in part, by the National Natural Science Foundation of China (no. 11161018), in part, by the NSF of Guangxi Zhuang Autonomous Region (no. 2012GXNSFAA053009), and, in part, by the Natural Science Foundation of Fujian Province of China (no. 2012J01014). The authors are grateful to the anonymous referees for their careful comments and valuable suggestions on this paper.

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