Discrete Dynamics in Nature and Society
Volume 2017 (2017), Article ID 8290906, 8 pages
https://doi.org/10.1155/2017/8290906

Research Article

Certain Nonlinear Integral Inequalities and Their Applications

Yazhou Tian,¹ Min Fan,² and Yuangong Sun¹

¹School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, China
²Qingdao Technological University, Feixian, Shandong 273400, China

Correspondence should be addressed to Yazhou Tian; tianyazhou369@163.com

Received 29 September 2016; Revised 5 December 2016; Accepted 26 December 2016; Published 19 January 2017

Academic Editor: Douglas R. Anderson

Copyright © 2017 Yazhou Tian et al. 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.

Several new retarded integral inequalities of Gronwall-Bellman-Pachpatte type are presented which generalize the inequalities to the more general nonlinear case in the literature and provide explicit bounds on unknown functions. These results include many existing ones as special cases and can be used as tools in the qualitative analysis of certain classes of integrodifferential equations.

1. Introduction

It is well known that integral inequalities are very useful to investigate the existence, uniqueness, boundedness, oscillation, and other qualitative properties of solutions to differential equations and integrodifferential equations [1–19]. One of the fundamental inequalities is the Gronwall inequality, which was established in 1919 by Gronwall [1]. As a generalization of Gronwall inequality, Gronwall-Bellman inequality plays a key role in studying stability and asymptotic behavior of solutions to differential equations and integrodifferential equations. An important nonlinear generalization of Gronwall-Bellman inequality is Bihari’s inequality [2]. In recent years, there has been much research activity concerning linear and nonlinear generalizations of Gronwall-Bellman-Bihari type inequalities. To mention a few, Pachpatte [3] presented a new integral inequality and studied the boundedness properties of some linear integrodifferential equations, one of which we give below for the convenience of the reader.

Theorem 1 (see Pachpatte [3]). Let , , and be a nonnegative constant. If

then

for .

Very recently, Abdeldaim and El-Deeb [4, 5] investigated the following integral inequalities which generalized the main results of [3].

Theorem 2 (see [4, Theorem  2.1]). Let and suppose that are increasing functions with and , for , and is a nonnegative constant. If

then

Theorem 3 (see [4, Theorem ] and [5, Theorem ]). Let and assume that are increasing functions with , , , and , for , and and are positive constants. If

then

where

and and stand for the inverses of the functions and , respectively.

Note that the function of inequalities established in [4, 5] satisfies ; that is, is global Lipchitz. However, in the real system there may exist the non-Lipschitz , such as or , and then the obtained results can not be applied to this kind of systems. The natural question now is as follows: is it possible to relax this condition? The aim of this paper is to give an affirmative answer to this question.

In this paper, we study certain classes of integral inequalities which generalize the inequalities established in [3–5] to the more general nonlinear case. The obtained results can be used as tools in the study of qualitative theory of certain classes of integrodifferential equations with more general nonlinearities. At the end, two examples are provided to illustrate the main results.

2. Main Results

In what follows, denotes the set of real numbers, and denote the sets of all continuous functions and all continuously differentiable functions defined on set with range in the set , respectively, and stands for the inverse to .

The following lemmas are very useful in the proof of our main results.

Lemma 4 (see [6, Lemma ]). Assume that and . Then

for any .

Lemma 5 (see [7, Corollary ]). Assume that , , and is a positive constant. If

then

where is the largest number such that

Theorem 6. Let , , , and be nonnegative constants satisfying , , , , , , and . If

then, for any ,

where

Proof. Define a function by

Then, , is nondecreasing, and

Differentiating and using (16), we conclude that

By virtue of Lemma 4, for any ,

Combining (17) and (18), we deduce that

Define another function by

Then is nondecreasing, , and

It follows from (19) that

Differentiating and using (22), we obtain

Integrating the latter inequality from to , we conclude that

Therefore,

which yields

By virtue of , (13) holds. This completes the proof.

Remark 7. If , then Theorem 6 reduces to Theorem 2.

Remark 8. If and , then Theorem 6 reduces to Theorem 1.

Theorem 9. Let . Assume further that , , , , , , , , , , , and is a positive constant. If, for ,

then

where

is the largest number such that

Proof. Define a function by

Then, , is nondecreasing, and

It is not difficult to obtain

where

is nondecreasing, , and . Differentiating , we have

Noting that is increasing and is nondecreasing, we get , and so

Integrating the latter inequality from to , we have

By virtue of Lemma 5,

This and (33) imply that

and thus

Integrating (40) from to ,

which yields (28) due to (32). The proof is complete.

Remark 10. The condition in [4, 5] is relaxed to , where and are nonnegative constants. On the basis of this new assumption, the classes of are enlarged. For example, satisfies when choosing and .

If we choose , , and in Theorem 9, then . Letting , , we have , and hence . Therefore, satisfies with and , and then we can obtain the following result.

Corollary 11. Assume that , , , , , is a positive constant, and, for ,

Then

where ,

and is the largest number such that

Proof. By virtue of and , we obtain and

An application of Theorem 9 implies that

This completes the proof.

3. Applications

In this section, we will use the inequalities established in Theorems 6 and 9 to study the boundedness of certain integrodifferential equations.

Example 1. Consider the following integral equation:

where is a continuous function defined on , , is a nonnegative constant, and and satisfy

where , , and are nonnegative constants satisfying , , and . Then every solution of (48) satisfies

Now, using Theorem 6, we obtain that, for any ,

where

which illustrates that the solution of (48) is global boundedness.

Example 2. Consider the integrodifferential equation

where is a constant, , and .
Assume that

where . Then the solution of (53) satisfies

and hence

A suitable application of Corollary 11 to (56) yields

where

and is the largest number such that

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (61174217, 61374074, 61473133, and 11671227) and the Doctoral Scientific Research Foundation of University of Jinan (1008398).

References

T. H. Gronwall, “Note on the derivatives with respect to a parameter of the solutions of a system of differential equations,” Annals of Mathematics. Second Series, vol. 20, no. 4, pp. 292–296, 1919.

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.

B. G. Pachpatte, Inequalities for Differential and Integral Equations, vol. 197 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1998.

A. Abdeldaim and A. A. El-Deeb, “On generalized of certain retarded nonlinear integral inequalities and its applications in retarded integro-differential equations,” Applied Mathematics and Computation, vol. 256, pp. 375–380, 2015.

A. Abdeldaim and A. A. El-Deeb, “On some generalizations of certain retarded nonlinear integral inequalities with iterated integrals and an application in retarded differential equation,” Journal of the Egyptian Mathematical Society, vol. 23, no. 3, pp. 470–475, 2015.

F. Jiang and F. Meng, “Explicit bounds on some new nonlinear integral inequalities with delay,” Journal of Computational and Applied Mathematics, vol. 205, no. 1, pp. 479–486, 2007.

W.-S. Wang, X. Zhou, and Z. Guo, “Some new retarded nonlinear integral inequalities and their applications in differential–integral equations,” Applied Mathematics and Computation, vol. 218, no. 21, pp. 10726–10736, 2012.

R. Bellman, “The stability of solutions of linear differential equations,” Duke Mathematical Journal, vol. 10, pp. 643–647, 1943.

Y. Tian, M. Fan, and F. Meng, “A generalization of retarded integral inequalities in two independent variables and their applications,” Applied Mathematics and Computation, vol. 221, pp. 239–248, 2013.

Y. Tian, Y. Cai, L. Li, and T. Li, “Some dynamic integral inequalities with mixed nonlinearities on time scales,” Journal of Inequalities and Applications, vol. 2015, article 12, pp. 1–10, 2015.

C. Zhang, T. Li, B. Sun, and E. Thandapani, “On the oscillation of higher-order half-linear delay differential equations,” Applied Mathematics Letters, vol. 24, no. 9, pp. 1618–1621, 2011.

W. N. Li and W. H. Sheng, “Some nonlinear dynamic inequalities on time scales,” Proceedings of the Indian Academy of Sciences-Mathematical Sciences, vol. 117, pp. 545–554, 2007.

B. Zheng, “Explicit bounds derived by some new inequalities and applications in fractional integral equations,” Journal of Inequalities and Applications, vol. 2014, article no. 4, 2014.

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.

O. Lipovan, “A retarded Gronwall-like inequality and its applications,” Journal of Mathematical Analysis and Applications, vol. 252, no. 1, pp. 389–401, 2000.

Y. Sun and T. Hassan, “Some nonlinear dynamic integral inequalities on time scales,” Applied Mathematics and Computation, vol. 220, pp. 221–225, 2013.

X. Liu, L. Zhang, P. Agarwal, and G. Wang, “On some new integral inequalities of Gronwall-Bellman-BIHari type with delay for discontinuous functions and their applications,” Indagationes Mathematicae, vol. 27, no. 1, pp. 1–10, 2016.

P. Agarwal and J. Choi, “Certain fractional integral inequalities associated with pathway fractional integral operators,” Bulletin of the Korean Mathematical Society, vol. 53, no. 1, pp. 181–193, 2016.

Q.-H. Ma and E.-H. Yang, “On some new nonlinear delay integral inequalities,” Journal of Mathematical Analysis and Applications, vol. 252, no. 2, pp. 864–878, 2000.