Research Article | Open Access
A Multiple Iterated Integral Inequality and Applications
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.
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  discussed In 1990 Pinto  investigated Replacing the upper limit of the integral with a function in (2), in 2000 Lipovan  improved Bihari’s results by investigating the following so-called retarded Gronwall-like inequalities: In 2005 Agarwal et al.  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  in 2004.
In 2008 Ma and Peari  discussed more generally the following inequality: where . In 2011 Abdeldaim and Yakout  investigated the following: In 2013 Wang et al.  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.
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 .
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.
Lemma 1. Suppose that are nonnegative and integrable on . Then are increasing and continuous differentiable functions, and
2.2. Simplification with Operators
Let be positive continuous functions in (11), where , . Define by and define by
Define a function
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
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).
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.
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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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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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.