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A New Nonlinear Retarded Integral Inequality and Its Application

Journal of Inequalities and Applications20102010:462163

Received: 28 April 2010

Accepted: 15 August 2010

Published: 18 August 2010


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.


Differential EquationContinuous FunctionIntegral EquationRecent WorkAsymptotic Behavior

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
Agarwal et al. [2] obtained the explicit bound to the unknown function of the following retarded integral inequality
Cheung [3] investigated the inequality in two variables
Chen et al. [4] discussed the following inequality in two variables
Pachpatte [8] obtained an upper bound in the following inequality:
Pachpatte [9] firstly got the estimation of the unknown function of the following inequality:
then, the estimation was used to study the boundedness, asymptotic behavior, slowly growth of the solutions of the integral equation

(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

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

Theorem 2.1.

Suppose that ( )–( ) hold and is a nonnegative and continuous function on satisfying (1.8). Then one has
for all , where
, and denote the inverse function of and , respectively, and is arbitrarily given on the boundary of the planar region


From the inequality (1.8), for all , we have
where is chosen arbitrarily, using the assumption . For convenience, we define a function by the right-hand side of (1.8), that is,
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
for all . From (2.7), we get
By taking in (2.8) and then integrating it from to , we get
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
where is defined by (2.4). From (2.10), we have
for all . By taking in (2.11) and then integrating it from to , using the definition of in (2.1), we get
Using the fact and , from (2.12) we obtain
Let , from (2.13)we observe that

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:

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


From the integral equation (3.1), we have

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.



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).

Authors’ Affiliations

Department of Mathematics, Hechi University, Guangxi, China
School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, China


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© Wu-Sheng Wang et al. 2010

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