Journal of Function Spaces
Volume 2021 (2021), Article ID 5519992, 7 pages
https://doi.org/10.1155/2021/5519992

Research Article

Existence Results for Fractional Semilinear Integrodifferential Equations of Mixed Type with Delay

Xue Wang1,2 and Bo Zhu²

¹School of Mathematics, Shandong University, Jinan 250100, China
²School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Jinan 250014, China

Correspondence should be addressed to Bo Zhu; zhubo207@163.com

Received 18 January 2021; Revised 9 February 2021; Accepted 23 February 2021; Published 2 March 2021

Academic Editor: Chuanjun Chen

Copyright © 2021 Xue Wang and Bo Zhu. 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.

In this paper, we discuss a class of fractional semilinear integrodifferential equations of mixed type with delay. Based on the theories of resolvent operators, the measure of noncompactness, and the fixed point theorems, we establish the existence and uniqueness of global mild solutions for the equations. An example is provided to illustrate the application of our main results.

1. Introduction

Fractional calculus can be used to describe some nonclassical phenomena in natural science and engineering applications. Fractional differential equations have been applied in different fields ranging from engineering, finance, and physics in the past few decades. Researchers have conducted extensive explorations on this subject and have achieved fruitful results for the fractional differential equations [1–13]. Zhu and Han [10] and Chadha and Pandey [11] studied the fractional integrodifferential equations and discussed the existence of mild solutions. Based on the theory of the resolvent family and fixed point theorems, Chen et al. [14–17] analyzed nonautonomous evolution equations in a Banach space. Moreover, some researchers considered sufficient conditions on the existence of mild solutions for fractional differential equations by the measure of noncompactness [4, 18, 19]. The initial boundary value problem for the fractional integrodifferential equations with delay has been investigated by using fixed point theorems [4, 5, 18, 20]. In [3, 21–24], differential equations of mixed type have been studied and some results have been concluded.

Chen [22] studied the fractional nonautonomous evolution equations of mixed type:

where

where the kernels and are linear functions. The operator is an integral with a variable upper limit, and the operator is an ordinary definite integral; accordingly, problem (1) is called fractional semilinear integrodifferential equations of mixed type.

Li and Jia [25] investigated the existence of mild solutions for abstract delay fractional differential equations:

where , is the Riemann-Liouville fractional integral, the linear operator is independent on , and the Lipschitz coefficient of is constant.

To the best of our knowledge, there are no results on the fractional integrodifferential equations of mixed type with delay. Motivated by this idea, we consider the following problem:

where , is the Caputo fractional derivative of order , is a closed and linear operator with domain defined on a Banach space , is the Riemann-Liouville fractional integral of order , and are defined by

where and are continuous and nonlinear functions, , , , , is to be specified later, and means the element of defined by , for .

We demonstrate the existence and uniqueness of global mild solutions for problem (4) under the conditions of the compact resolvent operator and noncompact resolvent operator, respectively. The kernels and of the operators and are nonlinear functions. In addition, the operator is dependent on The rest of this paper is organized as follows. Basic definitions and auxiliary results are presented in Section 2. In Section 3, we prove the existence and uniqueness of mild solutions via various fixed point theorems, the measure of noncompactness, and the Banach contraction mapping principle. An example is provided to illustrate the main theorems in Section 4. Finally, Section 5 is the summary of our results.

2. Preliminaries

Definition 1 [6, 26]. The Riemann-Liouville fractional integral and derivative of a function of order are defined by

where , denotes the gamma function, and .

Remark 2 [25]. , where and .

Definition 3 [26, 27]. The Caputo fractional derivative of order of a function is given by

Remark 4 [25]. For the Riemann-Liouville fractional integral operator and the Caputo fractional derivative operator, the following conclusions are obtained:

Definition 5 [28, 29]. Let be a closed and linear operator with domain defined on a Banach space and . Let be the resolvent set of . is called the generator of a -resolvent family if there exist and a strongly continuous function such that and

In this case, is called the -resolvent family generated by .

Remark 6 [29, 30]. satisfies the following properties:

(1) and , for (2) is strongly continuous for (3)If is compact for , then is continuous in the uniform operator topology

Lemma 7 [21]. Let be equicontinuous and bounded; then, is also equicontinuous and bounded.

Lemma 8 [24]. Let be equicontinuous and bounded; then, is continuous on and

where denotes the measure of noncompactness.

Lemma 9 [21]. Let be a Banach space and be bounded; then, there exists a countable set such that .

Lemma 10 [31]. Let be a Banach space and be a bounded closed and convex set. Assume that is a strict set contraction mapping; then, has at least one fixed point in .

Definition 11. A function is a mild solution of problem (4), if satisfies the following equations:

3. Main Results

Let us introduce the operator by

Theorem 12. Assume that the following conditions hold:
(H₁). The resolvent operator is compact for all .
(H₂). and are continuous; there exist nonnegative Lebesgue integrable functions such that and , for all .
(H₃). is continuous; there exist nonnegative Lebesgue integrable functions such that , for all .
Then, problem (4) has at least one mild solution .

Proof. Let us set the notation such that

where and .
First of all, we consider the set and show that . By using conditions (H₂) and (H₃), for all , we have

So, we conclude that maps into itself.
Second, we prove that is continuous.
Let , with . Using the fact that , , and are continuous, we obtain

for any uniformly. That is, for any , there exists a natural number , for , such that

which implies that

In consequence, is continuous.
Furthermore, we prove that is equicontinuous.
To do this, let . Obviously, it is a nonnegative Lebesgue integrable function. For all , we have

In view of condition , compactness of the resolvent operator implies the continuity in the uniform operator topology. That is, for any , there exists , for any , such that . Hence, for the above , by using properties of and the above inequalities, there exists () such that , for any. Consequently, is equicontinuous.
In the end, we prove that is precompact.
For any fixed and , the operator is defined by

Since is a compact resolvent operator, then the set is relatively compact in for any ().
Moreover, for any , one can find that

Thus, is totally bounded. Hence, is relatively compact in , and so, based on the Arzelà-Ascoli theorem, is completely continuous. As all the assumptions of the Schauder fixed point theorem are satisfied, the conclusion implies that the operator has a fixed point in , which is a global mild solution of problem (4). This completes the proof.

Next, we develop the existence of global mild solutions for problem (4) via the measure of noncompactness and fixed point theorem. Furthermore, we employ the notations: , , , and .

Theorem 13. Assume that (H₁) and the following conditions hold:
(H₄). The function is bounded and continuous, which satisfies

where .
(H₅). For any , there exist nonnegative Lebesgue integrable functions , such that for any equicontinuous and countable set , , , and .
(H₆).
Then, problem (4) has at least one mild solution.

Proof. By (H₄), there exists and , for any , such that

Let ; we first consider the set and show that . From the above inequality, for all , we have

Meanwhile, applying the arguments employed in the proof of Theorem 12, we conclude that is a continuous and bounded operator on .
Then, we prove that is equicontinuous. For any , we have

By (H₁), the compactness of , for , implies the continuity in the uniform operator topology. Namely, for any , there exists , for any , such that

Therefore, for the above , there exists () such that , for all , which shows that is equicontinuous. In view of Lemma 7, is bounded and equicontinuous.
Finally, we prove that is a condensing operator. By Lemma 9, for any , there exists a countable set such that

By using condition (H₅) and Lemma 8, we obtain

In addition, using Lemma 8, we have

Consequently,

By (H₆), we obtain that is a condensing operator on . By Lemma 10, there exists at least one fixed point for . In conclusion, problem (4) has at least one global mild solution. This completes the proof.

Remark 14. Theorems 12 and 13 above are concluded under the conditions that is compact for and the functions , , and satisfy corresponding conditions; in contrast, when the resolvent operator is noncompact, we could obtain Theorem 15 if , , and meet the Lipschitz conditions.

Theorem 15. Assume that the following conditions hold:
(H₇). is continuous; there exist nonnegative Lebesgue integrable functions , for all , such that

(H₈). and ; there exist nonnegative Lebesgue integrable functions , for all such that

(H₉).
Then, problem (4) has a unique mild solution.

Proof. For any ,

By (H₉), we have . These arguments enable us to conclude that the operator is a contraction mapping. Hence, the operator has a unique fixed point , which implies that problem (4) has a unique global mild solution. This completes the proof.

Remark 16. In Theorem 15, we develop the uniqueness of the mild solution for problem (4) via the Banach contraction mapping principle. In conditions (H₇) and (H₈), turn out to be nonnegative Lebesgue integrable functions instead of constants.

4. An Application

In order to show the application of the main results, we consider the following problem:

where , is the Caputo fractional derivative of order , is the Riemann-Liouville fractional integral of order , is a bounded domain with regular boundary , and

By setting , problem (33) can be rewritten as the following abstract form:

where , , and

It is well known that the operator generates a -resolvent family [23, 25]. Let equation (34) satisfy the conditions of Theorems 12–15; then, problem (34) has a global mild solution, which means that problem (33) has a mild solution.

5. Conclusion

In this paper, we study the existence and uniqueness of the global mild solutions for the fractional integrodifferential equations of mixed type with delay. Under the condition of the compact resolvent operator, we obtain Theorems 12 and 13, respectively, via various fixed point theorems and the measure of noncompactness. Theorem 15 is established by using the Banach contraction mapping principle under the condition of the noncompact resolvent operator. Furthermore, an example is provided to illustrate the main theorems. The kernels and of the operators and are nonlinear functions; meanwhile, the operator is dependent on . As a consequence, our main theorems improve and generalize many corresponding results by using different methods.

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 authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Acknowledgments

This research is supported by the National Science Foundation of China (Grant No. 11971264), the National Key R&D Program of China (Grant No. 2018YFA0703900), the National Natural Science Foundation of China (No. 62073190), and the Project of Shandong Province Higher Educational Science and Technology Program (No. J16LI14).

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