Existence of Solutions for a Periodic Boundary ValueProblem with Impulse and Fractional Derivative Dependence

Li, Yaohong; Wang, Yongqing; O’Regan, Donal; Xu, Jiafa

doi:https://doi.org/10.1155/2020/6625056

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Abstract Introduction Preliminaries Examples Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

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Volume 2020 | Article ID 6625056 | https://doi.org/10.1155/2020/6625056

Existence of Solutions for a Periodic Boundary ValueProblem with Impulse and Fractional Derivative Dependence

Yaohong Li,¹Yongqing Wang,²Donal O’Regan,³and Jiafa Xu⁴

Academic Editor: Xiaolong Qin

Received10 Oct 2020

Revised06 Dec 2020

Accepted09 Dec 2020

Published29 Dec 2020

Abstract

In this paper, we present some theorems on impulsive periodic boundary value problems with fractional derivative dependence. In particular, we discuss the existence of solutions of a class of fractional-order impulsive periodic boundary values with nonlinear terms and impulsive terms satisfying certain growth conditions. Three examples are provided to illustrate our results.

1. Introduction

This paper considers the existence of solutions of the following fractional-order impulsive periodic boundary value problem:where and represent the common Caputo derivatives of orders and , and , and . Here, and are continuous functions. Now, , where denote the right limit and the left limit of at the impulsive point . Also, , where denote the right limit and the left limit of at the impulsive point . If exist, we let . Also, are two real constants with .

The theory of fractional differential equation has received a lot of attention because of its wide application in mathematical models (see [1–27] and the references therein). Fractional-order impulsive differential equations are a natural generalization of the case of nonimpulses and are used to describe sudden changes in their states, such as in optimal control, population dynamics, biological systems, financial systems, and mechanical systems with impact. We refer the reader to [28–36] and the references therein. In particular, Bai et al. [37] investigated a mixed boundary value problem of nonlinear impulsive fractional differential equation:and some sufficient conditions on the existence and uniqueness of solutions for problem (2) are obtained under Lipschitz conditions. In [38], Zhang and Xu studied the following impulsive periodic boundary value problem with the Caputo fractional derivative:using Green’s function in [36], and via the symmetry property of Green’s function and topological degree theory, the authors obtained the existence of positive solutions for (3) when the growth of is superlinear and sublinear.

Inspired by the above research studies, in this paper, we consider fractional-order impulsive differential equations with generalized periodic boundary value conditions (1), where the nonlinear term, impulse terms, and periodic boundary conditions all depend on unknown functions and the lower-order fractional derivative of unknown functions. This is obviously more general and more widely applied, but it is also more complex and difficult to solve. Compared with (1), the nonlinear term, pulse term, and periodic boundary conditions of (3) are all independent of fractional derivatives, so it is a special form of (1). In this paper, we first give an equivalent integral form of solutions for problem (1) using some new Green’s functions. Next, we present some sufficient conditions for the existence of solutions for problem (1), where the nonlinear and impulse terms satisfy some nonlinear and linear growth conditions, which are different from the conditions in [36–38]. Finally, we present three examples to illustrate our main results.

2. Preliminaries and Lemmas

In this section, we only present some necessary definitions and lemmas about fractional calculus.

Definition 1. (see [39, 40]). The Riemann–Liouville fractional integral of order for a function is defined aswhere is the Euler gamma function.

Definition 2. (see [39, 40]). The Caputo fractional derivative of order for a continuous and n-order differentiable function is defined aswhere is the Euler gamma function and is the smallest integer greater than or equal to .

Lemma 1 (see [39, 40]). Let . The differential equation has a unique solution:for some , where .

Lemma 2. Let and . The unique solution of the following periodic boundary value problemis expressed bywhere

Furthermore,where

Proof. Suppose is a general solution of (7) on each interval . Then, using Lemma 1, (7) can be transformed into the following equivalent integral equation:where . Also, we haveFrom (12) and (13), according to (7), we obtainApplying the right fractional-order impulsive condition of (7), we obtainFrom (15) and (16), after a recursive calculation, we haveSimilar to (18), we see thatFrom (13), (14), and (16), we haveFrom (17) and (20), after a recursive calculation, we haveFor , substituting (18) and (20) into (12) and (13), we obtainwhere are defined by (7) and (9).
For , substituting (20) and (18) into (11) and (12), we havewhere are defined by (9) and (11). The proof is completed.

Lemma 3. Let . Then, and defined as in (9) and (11) are continuous, and the following inequalities hold:(i)(ii).

Proof. Directly observe thatLet , and exist, where . Note [35] that is a Banach space equipped with the norm

Lemma 4. If the function is continuous, then is a solution of (1) if and only if is a solution of the following integral equation:

Proof. Assume that satisfies (1). From Lemma 2, we see that satisfies integral equation (26).
Conversely, assume that satisfies integral equation (26). Applying Definition 2, by a direct fractional derivative computation, it follows that the solution given by (26) and (2) satisfies (1).
Define an operator asIt is easy to prove that the function is a solution of (1) if and only if is a fixed point of the operator .
For convenience, we list some hypotheses: (B1) with (B2) and are continuous functions

Lemma 5. Assume that (B1) and (B2) hold. Then, the operator defined as in (27) is completely continuous.

Proof. We divide the proof into three steps. Set for some . The steps are as follows:(i)Step 1. is continuous from the continuity of the functions .(ii)Step 2. is uniformly bounded. Now, for we have , where . In fact, for each , from Lemma 3, we have which and Lemma 4 imply that(iii)Step 3. is equicontinuous. For any , fixed and for any , there exists a constant such that for , we have Then, Thus, which implies that is equicontinuous on any subinterval . From the Arzela–Ascoli theorem, we deduce that is completely continuous.

Lemma 6 (Schauder fixed-point theorem, see [41, 42]). Let be a real Banach space, be a nonempty closed bounded and convex subset, and be compact. Then, has at least one fixed point in .

Lemma 7 (Krasnoselskii fixed point theorem, see [41, 42]). Let be a closed convex and nonempty subset of a Banach space . Let be the operators such that (i) whenever ; (ii) is compact and continuous; and (iii) is a contraction mapping. Then, there exists an such that .

Lemma 8 (Banach’s fixed point theorem, see [43]). Let be a Banach space, be closed, and be a strict contraction, i.e., for some and all . Then, has a unique fixed point in .

3. Existence of the Solutions

For convenience, we give the following symbols:

Now, we present our main theorems.

Theorem 1. Assume that (B1) and (B2) hold, and the following hypotheses are satisfied: (C1) There exist three nonnegative functions and two constants such that (C2) There exist eight positive constants and such that

Then, (1) has at least one solution in .

Proof. LetNow, is a closed bounded convex subset of .
For each , from (C1) and (C2), we haveFrom Lemma 3, we obtain thatwhich implies that .
From Lemmas 5 and 6, has at least one fixed point in , so (1) has at least one solution in .

Theorem 2. Assume that (B1) and (B2) hold, and the following hypotheses are satisfied: (C3) There exists a nonnegative function , such that (C4) There exist four positive constants such that If , then (1) has at least one solution in .

Proof. We first define the operators. For , letNow,LetLetNote that is a nonempty bounded closed convex subset of .
From Lemma 5, is completely continuous (i.e., condition (ii) of Lemma 7 is satisfied).
For any , from hypothesis (C4), we haveTherefore,and since , is a contraction (so condition (iii) of Lemma 7 is satisfied).
For each , from hypothesis (C3), we haveConsequently,For each , we havewhereThus, for any , we obtainwhich implies that (so condition (i) of Lemma 7 is satisfied).
In view of Lemma 7, there exists a such that , so (1) has at least one solution in .

Theorem 3. Assume that (B1), (B2), and (C4) hold and the following hypothesis is satisfied:(C5)There exist two nonnegative functions such that If , then (1) has a unique solution in .

Proof. ChoosewhereFirst, we show that , where . For , from hypotheses (C4) and (C5), we obtainThen,so .
Furthermore, from hypotheses (C4) and (C5), for all , we haveThus,where , so is a contraction. Lemma 8 guarantees that has a unique fixed point in , which is the unique solution of (1) in . This completes the proof.

4. Examples

In (1), let and then, we obtain the following fractional-order impulsive differential equation:

By a direct observation, note that with , so hypothesis (B1) is satisfied.

Example 1. In (60), letso hypothesis (B2) is satisfied. Set , and then, we obtainwhich implies that (C1) and (C2) are satisfied. Thus, all the hypotheses in Theorem 1 are satisfied, so (60) has at least one solution in .

Example 2. In (60), letso hypothesis (B2) is satisfied. Set , and then, we obtainwhich implies that (C3) and (C4) are satisfied. Also, note that . Then, all the hypotheses in Theorem 2 are satisfied, so (60) has at least one solution in E.

Example 3. In (60), letso hypothesis (B2) is satisfied. Set , and then, we obtainwhich implies that (C4) and (C5) are satisfied. Note that . Then, all the hypotheses in Theorem 3 are satisfied, so (60) has a unique solution in .

5. Conclusion

In this paper, we use fixed-point theorems to study fractional-order impulsive differential equation (1) with generalized periodic boundary value conditions. Very little is known on fractional-order impulsive differential equations with generalized periodic boundary value conditions where nonlinear terms and impulse terms depend on the unknown function and the lower-order fractional derivative of the unknown function. Our main results are obtained under some nonlinear and linear growth conditions corresponding to the relevant linear operators where the symmetry property of a Green’s function is not required, so our results generalize and improve works in the literature.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

The study was carried out in collaboration with all authors. All authors read and approved the final manuscript.

Acknowledgments

This work was supported by the University Natural Science Foundation of Anhui Provincial Education Department (Grant nos. KJ2019A0672 and KJ2018A0452), the Foundation of Suzhou University (Grant no. 2016XJGG13), the Natural Science Foundation of Chongqing (Grant no. cstc2020jcyj-msxmX0123), and the Technology Research Foundation of Chongqing Educational Committee (Grant nos. KJQN2019 00539 and KJQN202000528).

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Copyright © 2020 Yaohong Li 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.

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