Discrete Dynamics in Nature and Society
Volume 2018 (2018), Article ID 5841985, 8 pages
https://doi.org/10.1155/2018/5841985

Research Article

Some Nonlinear Delay Volterra–Fredholm Type Dynamic Integral Inequalities on Time Scales

Yazhou Tian,¹ A. A. El-Deeb,² and Fanwei Meng³

¹School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, China
²Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City, Cairo 11884, Egypt
³School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China

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

Received 22 February 2018; Accepted 25 July 2018; Published 12 August 2018

Academic Editor: Manuel De la Sen

Copyright © 2018 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.

We are devoted to studying a class of nonlinear delay Volterra–Fredholm type dynamic integral inequalities on time scales, which can provide explicit bounds on unknown functions. The obtained results can be utilized to investigate the qualitative theory of nonlinear delay Volterra–Fredholm type dynamic equations. An example is also presented to illustrate the theoretical results.

1. Introduction

Since Hilger established the theory of time scales [1], it has become the research focus of mathematics and engineering field [2]. Particularly, integral inequalities play an important role in studying the qualitative properties of dynamic equations on time scales. For example, the integral inequalities were employed to investigate the stability of switched systems or uncertain nonlinear systems [3, 4].

In recent years, many authors have been devoted to studying different kinds of integral inequalities and their applications [5–24], especially the application of Volterra–Fredholm integrodifferential system [25–28]. To mention a few, in [8], Gu and Meng considered the nonlinear dynamic integral inequalities on time scales and applied the theoretical results to Volterra–Fredholm integrodifferential system, and Liu [9] investigated the linear delay Volterra–Fredholm type dynamic integral inequalities which generalized the main results of [8]. In [20], Xu and Ma considered Volterra–Fredholm type integral inequalities in two independent variables and their applications in partial differential equations. Very recently, in [22], Ding and Ahmad studied Volterra–Fredholm type integral inequalities and their applications to fractional differential equations. As is known to us, few authors pay attention to nonlinear delay Volterra–Fredholm type dynamic integral inequalities on time scales. This is the main reason why we establish this topic.

This paper investigates a class of nonlinear dynamic integral inequalities on time scales, which can be utilized as effective tools in the study of delay Volterra–Fredholm type dynamic equations. At the end, we provide an example to illustrate the main results.

2. Main Results

Throughout the paper, let be the set of real numbers, , be the class of all continuous functions defined on set with range in the set , and be an arbitrary time scale. The set is derived from as follows: if has a left-scattered maximum , then ; otherwise, . , where , , and . denotes the set of rd-continuous functions. represents the set of all regressive and rd-continuous functions, and . The graininess function is defined by , the forward jump operator by , and the circle plus addition is defined by .

Next, we introduce some lemmas to establish the main results.

Lemma 1 (comparison theorem [2]). Let and . If

then

Lemma 2 (see [2]). Assume that is continuous at with and , and is rd-continuous on . For any , if there exists a neighborhood of , independent of , such that

where represents the derivative of with respect to , then

implies

Lemma 3 (see [6]). Let and . For any , we have

Theorem 4. Assume that , , and are positive constants, are rd-continuous functions, is nondecreasing, , , and . If satisfies

with the initial condition

then

under the condition that and , where

Proof. Define by

Then, , is nondecreasing,

and

Next, we will prove that, for every , .
Case 1. When , we obtain

Case 2. For with , by the initial condition (8), we have

From (14) and (15), we always have the relation

By simple computation, it follows from (12) and (16) that

For any , it follows from Lemma 3 that

This together with (17) implies

which yields

i.e.,

where . Note that and . By Lemma 1, we get

From (11) and (13), we get

i.e.,

Therefore,

Noting that , we obtain the required inequality (9). The proof is complete.

Remark 5. [8, Theorem 3] is the special case of Theorem 4 with , , , and .

Remark 6. Theorem 4 generalizes [9, Theorem 3.1] to the nonlinear case.

Remark 7. From (9), we can obtain that the upper bound of depends on parameters , and . For practical system, the above parameters are easily to be obtained; therefore the theoretical upper bound can be computed by (9).

If we take , then the following result can be obtained.

Corollary 8. Let be defined as in Theorem 4 and satisfy

and (8). If and , then

where are defined as in Theorem 4 and

Theorem 9. Suppose that , , and are positive constants, are rd-continuous functions, is nondecreasing, are defined as in Lemma 2 such that and for , , , and . If satisfies

and (8), then

under the condition that and , where

Proof. Define by

Then, for , is nondecreasing,

and

It is not difficult to obtain that

Taking the derivative of and by Lemma 2, we get

It follows from (33) and (35) that

By Lemma 3, we get

for any . Combining (37) with (38) yields

The remainder of the proof is similar to that of Theorem 4, and hence we omit it here.

Remark 10. In [14], we investigated the integral inequalities with mixed nonlinearities on time scales; however, the delay terms are not considered; furthermore, the inequalities considered in this paper are Volterra–Fredholm type. The results established in Theorems 4 and 9 generalize [14, Theorems 2.1 and 2.2].

3. Application

In this part, we present an example to illustrate the theoretical results.

Example 11. Consider the delay Volterra–Fredholm type integral equation on time scales

with the initial condition

where and defined on the interval are the state and state delay variables, respectively. , is nondecreasing, , , and . satisfying , , and , , and are continuous functions.

Suppose that

where and are real constants and , , and are nonnegative rd-continuous functions on . If and , then the solution of (40) satisfies

where

Actually, by (40), satisfies

It is not difficult to verify (43) is satisfied by Corollary 8.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All three authors contributed equally to this work. They all read and approved the final version of the manuscript.

Acknowledgments

This work was supported by the National Natural Science Foundations of China (11671227, 61703180), the Natural Science Foundation of Shandong Province (ZR2017LF012), A Project of Shandong Province Higher Educational Science and Technology Program (J17KA157), and the Doctoral Scientific Research Foundation of University of Jinan (1008398).

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