Journal of Applied Mathematics

Volume 2014 (2014), Article ID 327943, 9 pages

http://dx.doi.org/10.1155/2014/327943

Research Article

A Multiple Iterated Integral Inequality and Applications

Zongyi Hou and Wu-Sheng Wang

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

Received 11 June 2014; Accepted 10 July 2014; Published 22 July 2014

Academic Editor: Daoyi Xu

Copyright © 2014 Zongyi Hou and Wu-Sheng Wang. 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 new multiple iterated Volterra-Fredholm type integral inequalities, where the composite function of the unknown function with nonlinear function in integral functions in [Ma, QH, Pečarić, J: Estimates on solutions of some new nonlinear retarded Volterra-Fredholm type integral inequalities. Nonlinear Anal. 69 (2008) 393–407] is changed into the composite functions of the unknown function with different nonlinear functions , respectively. By adopting novel analysis techniques, the upper bounds of the embedded unknown functions are estimated explicitly. The derived results can be applied in the study of solutions of ordinary differential equations and integral equations.

1. Introduction

The well-known Gronwall-Bellman inequality [1, 2] is the following or can be equivalently regarded as the following:

where is a constant, is a given nonnegative continuous function, and is the unknown function. It is often used to estimate solutions of differential equations. In 1956 Bihari [3] discussed

In 1990 Pinto [4] investigated

Replacing the upper limit of the integral with a function in (2), in 2000 Lipovan [5] improved Bihari’s results by investigating the following so-called retarded Gronwall-like inequalities:

In 2005 Agarwal et al. [6] generally discussed

As required in estimation for solutions, invariant sets, and stability, many generalized versions of the Gronwall-Bellman inequality were given with an invariant decomposition [7–9], a singular kernel [10, 11], and maxima [12, 13]. More results about integral inequalities of single variable and multivariables can be found, for example, the books [14, 15].

In order to investigate the behavior of solutions of a linear Volterra-Fredholm type integral equation, a form of integral inequalities which contains multiple integrals of the unknown,

called linear Volterra-Fredholm type integral inequality with retardation, is discussed by Pachpatte [16] in 2004.

In 2008 Ma and Peari [17] discussed more generally the following inequality:

where . In 2011 Abdeldaim and Yakout [18] investigated the following:

In 2013 Wang et al. [19] studied a new integral inequality of Gronwall-Bellman-Pachpatte type

In this paper, on the basis of [17, 18], we discuss a new multiple iterated Volterra-Fredholm type integral inequality

Using monotonization of some functions, we simplify the above multicomposition in an operator form. The unknown function will be estimated by known functions. Furthermore, we apply our result to retarded nonlinear Volterra-Fredholm type equations for estimation of solutions.

2. Preliminaries

Throughout this paper, let denote the set of real numbers, and . For , let denote the class of th order continuously differentiable functions defined on the set and ranged in the set . For simplicity, we use the product and to present the composition and .

2.1. Monotonization

First, we monotonize those s in inequality (11). Define

recursively. One can prove that

(P1) each is a nondecreasing nonnegative continuous function,(P2), ,(P3) has stronger monotonicity than , denoted by , ; that is, by the definition given in [4, 6], the ratios , , are all nondecreasing.

Thus, the sequence can be replaced by a larger but monotonous one in (11). For a given constant , define functions , , recursively by

where we use and to denote and its inverse when there is no confusion. Clearly, they are all strictly increasing.

For given positive constants , , define by

where are defined by (13) and (14), respectively.

Lemma 1. Suppose that are nonnegative and integrable on . Then are increasing and continuous differentiable functions, and

Proof. By the definition, are increasing and continuous differentiable functions. From (13) and (14), we have

Moreover,

By the definitions of , we have the relation (17).

2.2. Simplification with Operators

Let be positive continuous functions in (11), where , . Define by

and define by

Having defined those operators, we can enlarge inequality (11) by (12) in the simpler form

where denotes a zero function.

2.3. -Function

Define by

where and are defined by (14) and (20), respectively. Since are continuous functions, are also continuous functions.

Define a function

Lemma 2. Suppose that are all continuous such that for , defined by (13) and (14) satisfy . Suppose that , and satisfy

where

and are defined by (23). Then, is nondecreasing, and has a solution with .

Proof. Using Lemma 1 and condition (25), we have

Thus is a nondecreasing function. Since and , we see that has a solution with .

3. Main Result

The following theorem shows that the unknown function is estimated by the given known functions.

Theorem 3. Let be a positive constant. Suppose that , , and , . Suppose that , , , and satisfy the assumption of Lemma 2. Suppose that is nondecreasing such that on . Then the unknown in (22) is estimated as

Remark 4. As explained in Remark 2 in [6], different choices of in the definitions (13)-(14) of do not affect our results (28).

Proof. For convenience, we cite some definitions in the discussion of our proof as follows: for each fixed positive continuous function , define by

where , , .

From (22), we have

Define a function by the function on the right-hand side of (30). Then, is a positive and nondecreasing function on . Using (30), we have

Differentiating with respect to , using (31) we have

for all . From (33), we have

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

for .

Let denote the function on the right-hand side of (35); we can see that is a positive and nondecreasing function on . From (35), we obtain

Differentiating with respect to , using (36) we obtain

From (38), we have

for all . From (39), we have

for all . Proceeding with the same derivation as in (36) to (40) and so on, we obtain

for all , where is defined by (14).

Define a function by the function on the right-hand side of (41). Then is a positive and nondecreasing function on . From (41), we get

Differentiating with respect to , we have

for all . Then (44) is equivalent to

for all . Integrating both sides of (45) from to , we have

for all .

Define a function by the function on the right-hand side of (46); then, is a positive and nondecreasing function on . From (46), we have

Differentiating with respect to , using (47) we have

for all . From (49), we have

for all , where denote the constant function . Integrating both sides of the above inequality from to , we obtain

for all . From (36), (42), (47), and (51), we have

for all . Substituting (37), (43), and (48) into (52), we have

for all . Since is chosen arbitrarily, we have

for all . By the definition of and (32), we have

From (54) and (55), we have

or

By the definition of , the assumption of Theorem 3, and (57), we observe that

By Lemma 2, is increasing. From the last inequality and (31) we have the desired estimation (28).

We define the following functions:

for all , where , are defined by (13) and (14), respectively.

Example 5. Let , , , , , , , be as in Theorem 3; is a positive constant. Suppose that the function is increasing and has a solution for . If satisfies (22), then

where are inverse functions of , respectively.

Remark 6. If in Example 5, then the result in Example 5 will yield the conclusion that appeared in Theorem in [17]. Since if , then , , from (60), we have for all .

4. Application

In this section, we apply our result in Theorem 3 to investigate the retarded Volterra-Fredholm integral equations

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

The following corollary gives the bound on the solution of (61).

Corollary 7. Suppose that the , in (61) satisfy the conditions

where , , , , and are as in Theorem 3; let . Assume that the function

is increasing and has a solution for . If is a solution of (61), then

where , , , and are as in Theorem 3.

Proof. Using the conditions (62)-(63) we have

for , where several changes of variables are made. Applying the result of Theorem 3 to the last inequality, we obtain the desired estimation (65).

Conflict of Interests

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

Acknowledgments

The authors are very grateful to the editor and the referees for their helpful comments and valuable suggestions. This research was supported by National Natural Science Foundation of China (Project no. 11161018) and the NSF of Guangxi Zhuang Autonomous Region (no. 2012GXNSFAA053009).

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