#### 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).