Research Article | Open Access

Yusong Lu, Wu-Sheng Wang, Xiaoliang Zhou, Yong Huang, "Generalized Nonlinear Volterra-Fredholm Type Integral Inequality with Two Variables", *Journal of Applied Mathematics*, vol. 2014, Article ID 359280, 14 pages, 2014. https://doi.org/10.1155/2014/359280

# Generalized Nonlinear Volterra-Fredholm Type Integral Inequality with Two Variables

**Academic Editor:**Maoan Han

#### Abstract

We establish a class of new nonlinear retarded Volterra-Fredholm type integral inequalities, with two variables, where known function in integral functions in Q.-H. Ma and J. Pečarić, 2008 is changed into the functions . By adopting novel analysis techniques, such as change of variable, amplification method, differential and integration, inverse function, and the dialectical relationship between constants and variables, the upper bounds of the embedded unknown functions are estimated. The derived results can be applied in the study of solutions of ordinary differential equations and integral equations.

#### 1. Introduction

Gronwall-Bellman inequality [1, 2] is an important tool in the study of existence, uniqueness, boundedness, oscillation, stability, and other qualitative properties of solutions of differential equations and integral equation. There can be found a lot of its generalizations in various cases from the literature (e.g., [3–6]).

Gronwall-Bellman inequality [1, 2] can be stated as follows. If and are nonnegative continuous functions on an interval satisfying for some constant , then During the past few years, some investigators have established a lot of useful and interesting integral inequalities in order to achieve various goals; see [7–25] and the references cited therein.

In 2004, Pachpatte [8] has established the linear Volterra-Fredholm type integral inequality with retardation In 2005, Agarwal et al. [9] investigated the inequality In 2006, Cheung [10] studied the inequality In 2008, Ma and Pečarić [13] have discussed the following useful nonlinear Volterra-Fredholm type integral inequality with retardation

In 2011, Abdeldaim and Yakout [20] studied a new integral inequality of Gronwall-Bellman-Pachpatte type

In this paper, on the basis of [13, 20], we discuss a new retarded nonlinear Volterra-Fredholm type integral inequality

where is a constant, where the known function in integral functions in [13] is replaced to the functions . The upper bound estimation of the unknown function is given by integral inequality technique. Furthermore, we apply our result to retarded nonlinear Volterra-Fredholm type equations for estimation.

#### 2. Main Result

Throughout this paper, and are the given subsets 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 .

Theorem 1. *Suppose that is a constant; functions , (), both and are nondecreasing with on , on , are nondecreasing functions with () for ,
**
is increasing, and has a solution for . If satisfies (8), then**where () are inverse functions of , respectively.*

*Proof. *Let denote the function on the right-hand side of (8), which is positive and nondecreasing in each of the variables . From (8), we have
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 all , , , and are chosen arbitrarily, where is defined by (9).

Let denote the function on the right-hand side of (18), which is positive and nondecreasing in each of the variables . From (18), we have
Differentiating with respect to , by the monotonicity of , , , and , the property of , and (19), we have
for all . From (21), we have

for all . From (22), we have

for all , where is defined by (10). Let denote the function on the right-hand side of (23), which is positive and nondecreasing in each of the variables . Then (23) is equivalent to

Differentiating with respect to , using (24), we have

for all . From (26), using the monotonicity of , , , and and the property of , we have

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

where is defined by (11).

From (19), (24), and (28), we haveSubstituting (20) and (25) into (29), we have Since are chosen arbitrarily, we have

By the definition of and (15), we have

or

By the definition of , the assumption of Theorem 1, and (34), we observe that Since is increasing, from (14), (31), and (35), we have the desired estimation (13).

We consider a special case of Theorem 1. If satisfies nonlinear Volterra-Fredholm type integral inequality with retardation,

Corollary 2. *Let functions , , , , , , , , , , and and constant be as in Theorem 1. Suppose that
**
is increasing and has a solution for . If satisfies (36), then
**for all , where () are inverse functions of , respectively.*

*Remark 3. *In Corollary 2, when on , , and , (38) is equivalent to

Corollary 2 reduces to Theorem in [13].

#### 3. Application

In this section, we apply our result in Theorem 1 to study the retarded Volterra-Fredholm integral equations with two variables, which demonstrates that our result is useful to investigate the qualitative properties of solutions of some retarded Volterra-Fredholm integral equations with two variables.

We consider the retarded Volterra-Fredholm integral equation of the form

for all , where , , and are strictly increasing with , , , , , , and . Let , ; then satisfy the condition in Theorem 1 and are invertible functions.

The following corollary gives the bound on the solution of (40).

Corollary 4. *Let . Suppose that in (40) satisfy the conditions
**
where , , , , , and are as in Theorem 1. Assume that the function**
is increasing and has a solution for . If is a solution of (40) on , then*