Boundedness of the Solutions to Certain Delay Dynamic Integrodifferential Systems on Time Scales

Abstract

In this paper, the bounds on the solutions of certain delay dynamic integrodifferential systems on time scales are considered. Based on a new Gronwall-Bellman type delay integral inequality, we can estimate the boundedness of solutions to integrodifferential systems. At the end, an example is presented to state the main results.

1. Introduction

The theory of time scales was established and developed by Hilger [1] and Bohner and Peterson [2, 3]. At present, different kinds of integral inequalities and their applications in differential, integral, and integrodifferential equations have become the research focus; see the papers [4–23]. To list a few, Ma and Pečarić [10] established an integral inequality on time scales to study the boundedness of solutions of the delay dynamic differential system. Wang and Xu [13] investigated some integral inequalities in two independent variables on time scales. In [15, 16], Ma et al. considered the generalized two-dimensional fractional differential system with Hadamard derivative. However, to the best of our knowledge, there are very little known results on discussing the bounds on the solutions of delay integrodifferential system on time scales.

Motivated by the works in [10, 15, 16], we further investigate the delay dynamic integrodifferential system on time scales. By introducing a new Gronwall-Bellman type integral inequality, we obtain the bounds on the solutions of a class of delay dynamic integrodifferential system on time scales.

In the following, denotes the set of real numbers and , represents the class of all continuous functions defined on set with range in the set . denotes an arbitrary time scale, , , and denotes the set of rd-continuous functions.

2. Problem Description and Preliminaries

Consider the following delay integrodifferential systemwith the initial conditionwhere , are continuous functions, , , , , are some positive constants, and .

Remark 1. If we let , the system considered in this paper reduces to the one in [10].
The following lemmas are useful in our main results.

Lemma 2 (see [4], Lemma 2.1). Assume that and . Thenfor any .

Lemma 3. Assume that , , and is nondecreasing. If thenwhere

Proof. First we assume that ; from (3), we have Defining a function by right side of (6), we obtain is nondecreasing, and which implies that where is defined as in (5). Combining (7) and (9), we get the required inequality (4).
If for , we carry out the above procedure with instead of , where is an arbitrary small constant, and subsequently pass to the limit as to obtain (4). The proof is complete.

3. Main Results

Theorem 4. Assume that hold, where (), (), is a continuous function satisfying for , where is continuous , and are constants, and is a constant with , . Furthermore, suppose that is a solution of system (1) satisfying the initial condition . Then, for any constant , where

Proof. The solution of system (1) satisfies Combining (10) and (14), we have Defining and we can obtain that and are nondecreasing, and Case 1: for with , we obtain Case 2: for with , by the initial condition , we have Both (19) and (20) imply that This together with (16) and (17) yields Define Combining (22) and (23), we have By Lemma 2 and (24), for any real number , we get andBy assumption (11) and the last inequalities, it follows that andSubstituting the last inequalities into (23), we can obtain andwhere and are defined as in (13).
Let From (30), we have Since is nondecreasing, by Lemma 3, we have where is defined as in (13). Since is nondecreasing, it follows from (33) that where is defined as in (13). Substituting (34) into (29), we can obtain Using Lemma 3, we get where and are defined as in (13). From (34), we have Combining (36) and (37) with (24) and (18), we obtain the desired inequality (12). This completes the proof.

Remark 5. Theorem 4 generalizes [10, Theorem 3.1].

Remark 6. Assumptions (10) and (11) are easily satisfied. For example, we can choose , and it is not difficult to verify that where .

4. Application

In this section, we present an example to illustrate the main results.

Example 7. Consider system (1) with the initial condition and , , , , and satisfy with . Then the solution of system (1) satisfies where In fact, the solution of system (1) satisfies the following integral equation Therefore, We can choose and Since can be selected as . Similarly, we choose Applying Theorem 4 to (42) yields (40).

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

Both the authors contributed equally to this work. They both read and approved the final version of the manuscript.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant nos. 61807015, 61473133), the Natural Science Foundation of Shandong Province (Grant no. ZR2017LF012), A Project of Shandong Province Higher Educational Science and Technology Program (Grant no. J17KA157), and the Doctoral Scientific Research Foundation of University of Jinan (Grant no. 1008398).