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A New Nonlinear Retarded Integral Inequality and Its Application

Abstract

The main objective of this paper is to establish a new retarded nonlinear integral inequality with two variables, which provide explicit bound on unknown function. This inequality given here can be used as tool in the study of integral equations.

1. Introduction

Being important tools in the study of differential equations, integral equations and integro-differential equations, various generalizations of Gronwall inequality and their applications have attracted great interests of many mathematicians. Some recent works can be found, for example, in [17] and some references therein. Agarwal et al. [1] studied the inequality

(1.1)

Agarwal et al. [2] obtained the explicit bound to the unknown function of the following retarded integral inequality

(1.2)

Cheung [3] investigated the inequality in two variables

(1.3)

Chen et al. [4] discussed the following inequality in two variables

(1.4)

Pachpatte [8] obtained an upper bound in the following inequality:

(1.5)

Pachpatte [9] firstly got the estimation of the unknown function of the following inequality:

(1.6)

then, the estimation was used to study the boundedness, asymptotic behavior, slowly growth of the solutions of the integral equation

(1.7)

(1.7) was studied by Gripenberg in [10].

However, the bound given on such inequality in [8] is not directly applicable in the study of certain retarded differential and integral equations. It is desirable to establish new inequalities of the above type, which can be used more effectively in the study of certain classes of retarded differential and integral equations.

In this paper, we establish a new integral inequality

(1.8)

We will prove importance of (1.8) in achieving a desired goal.

2. Main Result

Throughout this paper, are given numbers, and . For functions ,   denotes the derivative of , and denotes the partial derivative on . Consider (1.8), and suppose that

() is a strictly increasing function with and as ;

() are nondecreasing in each variable;

() are nondecreasing with for ;

() and are nondecreasing such that and ;

.

We define functions , and by

(2.1)

Theorem 2.1.

Suppose that ()–() hold and is a nonnegative and continuous function on satisfying (1.8). Then one has

(2.2)

for all , where

(2.3)

, and denote the inverse function of and , respectively, and is arbitrarily given on the boundary of the planar region

(2.4)

Proof.

From the inequality (1.8), for all , we have

(2.5)

where is chosen arbitrarily, using the assumption . For convenience, we define a function by the right-hand side of (1.8), that is,

(2.6)

By the assumptions ()–(), is a positive and nondecreasing function in each variable, . Differentiating both sides of (2.6) and using the fact that , we obtain

(2.7)

for all . From (2.7), we get

(2.8)

By taking in (2.8) and then integrating it from to , we get

(2.9)

for all , where using the definition of in (2.1). Similarly to the above statement, we define a function by the right-hand side of (2.9), then is a positive and nondecreasing function in each variable, and . Differentiating for , by the relation among and , we have

(2.10)

where is defined by (2.4). From (2.10), we have

(2.11)

for all . By taking in (2.11) and then integrating it from to , using the definition of in (2.1), we get

(2.12)

Using the fact and , from (2.12) we obtain

(2.13)

Let , from (2.13)we observe that

(2.14)

Since is arbitrary, from (2.14), we get the required estimation (2.2).

3. Applications

In this section, we present an application of our result to obtain bound of the solution of a integral equation:

(3.1)

where is a strictly increasing function with and as , is a given positive constant, are bounded functions and nondecreasing in each variable, functions and satisfy hypothesis , i=1,2, and is nondecreasing on such that for .

The integral equation (3.1) is obviously more general than (1.7) considered in [10]. When keeping fixed, let , then integral equation (3.1) reduces to integral equation (1.7) in [10].

Corollary 3.1.

Consider integral equation (3.1) and suppose that where . Then all solutions of (3.1) have the estimate

(3.2)

for all , where

(3.3)

Functions are defined as in Theorem 2.1, and is arbitrarily given on the boundary of the planar region

(3.4)

Proof.

From the integral equation (3.1), we have

(3.5)

Clearly, inequality (3.5) is in the form of (1.8). Thus, the estimate (3.2) of the solution in this corollary is obtained immediately by our Theorem 2.1.

References

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Acknowledgments

The authors are very grateful to the editor and the referees for their helpful comments and valuable suggestions. This paper is supported by the Natural Science Foundation of Guangxi Autonomous Region (0991265), the Scientific Research Foundation of the Education Department of Guangxi Autonomous Region (200707MS112), and the Key Discipline of Applied Mathematics of Hechi University of China (200725).

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Correspondence to Wu-Sheng Wang.

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Wang, WS., Luo, RC. & Li, Z. A New Nonlinear Retarded Integral Inequality and Its Application. J Inequal Appl 2010, 462163 (2010). https://doi.org/10.1155/2010/462163

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