About this Journal Submit a Manuscript Table of Contents
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

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 [79], 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 Pe ari [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).

References

  1. T. H. Gronwall, “Note on the derivatives with respect to a parameter of the solutions of a system of differential equations,” Annals of Mathematics, vol. 20, no. 4, pp. 292–296, 1919. View at Publisher · View at Google Scholar · View at MathSciNet
  2. R. Bellman, “The stability of solutions of linear differential equations,” Duke Mathematical Journal, vol. 10, pp. 643–647, 1943. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. I. Bihari, “A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations,” Acta Mathematica Academiae Scientiarum Hungaricae, vol. 7, pp. 81–94, 1956. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  4. M. Pinto, “Integral inequalities of Bihari-type and applications,” Funkcialaj Ekvacioj, vol. 33, no. 3, pp. 387–403, 1990. View at MathSciNet
  5. O. Lipovan, “A retarded Gronwall-like inequality and its applications,” Journal of Mathematical Analysis and Applications, vol. 252, no. 1, pp. 389–401, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  6. R. P. Agarwal, S. Deng, and W. Zhang, “Generalization of a retarded Gronwall-like inequality and its applications,” Applied Mathematics and Computation, vol. 165, no. 3, pp. 599–612, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  7. W. N. Zhang, “Projected Gronwall's inequality,” Journal of Mathematical Research and Exposition, vol. 17, no. 2, pp. 257–260, 1997 (Chinese). View at MathSciNet
  8. W. Zhang and S. Deng, “Projected Gronwall-Bellman's inequality for integrable functions,” Mathematical and Computer Modelling, vol. 34, no. 3-4, pp. 393–402, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. L. Zhou, K. Lu, and W. Zhang, “Roughness of tempered exponential dichotomies for infinite-dimensional random difference equations,” Journal of Differential Equations, vol. 254, no. 9, pp. 4024–4046, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  10. Q. H. Ma and E. H. Yang, “Estimates on solutions of some weakly singular Volterra integral inequalities,” Acta Mathematicae Applicatae Sinica, vol. 25, no. 3, pp. 505–515, 2002. View at MathSciNet
  11. Q.-H. Ma and J. Pečarić, “Some new explicit bounds for weakly singular integral inequalities with applications to fractional differential and integral equations,” Journal of Mathematical Analysis and Applications, vol. 341, no. 2, pp. 894–905, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. J. Henderson and S. Hristova, “Nonlinear integral inequalities involving maxima of unknown scalar functions,” Mathematical and Computer Modelling, vol. 53, no. 5-6, pp. 871–882, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  13. Y. Yan, “Nonlinear Gronwall-Bellman type integral inequalities with maxima,” Mathematical Inequalities & Applications, vol. 16, no. 3, pp. 911–928, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  14. D. Bainov and P. Simeonov, Integral Inequalities and Applications, vol. 57 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, The Netherlands, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  15. B. G. Pachpatte, Inequalities for Differential and Integral Equations, Academic Press, London, UK, 1998. View at MathSciNet
  16. B. G. Pachpatte, “Explicit bound on a retarded integral inequality,” Mathematical Inequalities & Applications, vol. 7, no. 1, pp. 7–11, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. Q. H. Ma and J. Pečarić, “Estimates on solutions of some new nonlinear retarded Volterra-Fredholm type integral inequalities,” Nonlinear Analysis: Theory, Methods &Applications, vol. 69, no. 2, pp. 393–407, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. A. Abdeldaim and M. Yakout, “On some new integral inequalities of Gronwall-Bellman-Pachpatte type,” Applied Mathematics and Computation, vol. 217, no. 20, pp. 7887–7899, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  19. W. Wang, D. Huang, and X. Li, “Generalized retarded nonlinear integral inequalities involving iterated integrals and an application,” Journal of Inequalities and Applications, vol. 2013, article 376, 2013. View at Publisher · View at Google Scholar · View at MathSciNet