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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

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., [37]). During the past few years, some investigators have established a lot of useful and interesting integral inequalities in order to achieve various goals; see [818] 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 ,