Existence and Uniqueness Theorems for a Fractional Differential Equation with Impulsive Effect under Band-Like Integral Boundary Conditions

Abstract

In this paper, we consider a class of nonlinear Caputo fractional differential equations with impulsive effect under multiple band-like integral boundary conditions. By constructing an available completely continuous operator, we establish some criteria for judging the existence and uniqueness of solutions. Finally, an example is presented to demonstrate our main results.

1. Introduction

Researches on fractional differential equations have witnessed an unprecedented boom in recent years on account of the far-reaching application in various subjects, such as physics, biology, nuclear dynamics, chemistry, etc., for more details, see [1–3] and the references therein. Considering the impulse effect in the continuous differential equation can quantify the impact of the instantaneous mutation of the model and provide a theoretical basis for the practical application. Therefore, impulsive differential equation problems also attract great attention from scholars. For the theories of impulsive differential equations, the readers can refer to [4–7]. In addition, there have been some excellent results concerning the existence, uniqueness, and multiplicity of solutions or positive solutions to some nonlinear fractional differential equations with various nonlocal boundary conditions. As for some recent bibliographies, we refer readers to see [8–11] and the reference therein.

Yang and Zhang in [12] studied the following impulsive fractional differential equation

where is the Caputo fractional derivative, . is a continuous function, are continuous functions, , . By transforming the boundary value problem into an equivalent integral equation and employing some fixed point theorems, existence result is obtained.

The research results of fractional differential equations with integral boundary conditions are also quite rich, and the research on those questions remains as a hotpot among many scholars in recent years. We refer readers to see [13–16] and the reference therein.

In [13], Song and Bai considered the following boundary value problem of fractional differential equation with Riemann–Stieltjes integral boundary condition

where is the Riemann–Liouville fractional derivative, is a function of bounded variation, denotes the Riemann–Stieltjes integral of with respect to . By the use of fixed point theorem and the properties of mixed monotone operator theory, the existence and uniqueness of positive solutions for the problem are acquired.

Moreover, Zhao and Liang in [14] added impulsive effect to fractional equations with integral boundary conditions and discussed the existence of solutions

where is the Riemann-iouville fractional derivative of order . By applying the contraction mapping principle and the fixed point theorem, some sufficient criteria for the existence of solutions are obtained.

Inspired by the works above, we will study the impulsive fractional differential equation with band-like integral boundary conditions

where , are the Caputo fractional derivatives of order , for , and is a nonnegative constant, satisfying , , and represent the right and the left limits of at . By using the Leray–Schauder alternative theorem and the Banach contraction mapping principle, the existence and uniqueness theorems of solutions to problem (4) can be established.

We emphasize that the discontinuous points caused by impulse are just the upper and lower limits of the band-like integral values in the boundary conditions of (4). In other words, the value of the unknown function at the endpoint of the interval [0,1] is related to the linear combination of the integral values of the unknown function between the discontinuous points.

Another thing worth mentioning is that despite the complicated boundary conditions and the interference of the impulse, we use a piecewise function to represent the operator in a concise form based on the form of the Green’s function and accurately estimate the upper bound of its absolute value, which is fully prepared for the establishment of the main theorem.

Accordingly, the conclusions we reached are extensive results compared with the reference [4–7, 15–20] and a meaningful supplement to the theory of impulsive fractional differential equations.

2. Preliminaries

In this section, we present some definitions, lemmas, and some prerequisite results that will be used to prove our results.

Definition 1 [19]. The Riemann–Liouville fractional integral of order of a function is defined asif the right-hand side is pointwise defined on where is the Euler gamma function satisfying for

Definition 2 [16]. The Caputo fractional derivative of order for a function is defined aswhere and stands for the largest integer that not greater than .

Lemma 1. For the solution of the fractional differential equation can be expressed aswhere for .

Lemma 2. For any , the following boundary value problemhas a unique solutionwherewhere

Proof. From equation (9), through calculation we havewhere is an arbitrary real constant.
For , according to (15), we can obtainwhere is an arbitrary real constant.
For , based on (15) and (16), we haveAnalogously, for , it holds thatSince , together with (16) and (18), we receive thatSubstituting into (18), and based on the form of Green’s function, we getSubject to (20), using boundary conditions of (10), we haveConsequently,where In what follows, we always assume that
From (18)–(20), and (22), it can be received thatwhere are denoted by (12).
Define exit and . Obviously, is a Banach space endowed with the norm .

Lemma 3. For any , the following results are true
(1), for (2) for (3)(4)

Proof. For , we haveAccording to (24) and (25), we getand

Apparently, , for and so in view of Lemma 3, and combing with (11), we can write

Lemma 4 (the Leray–Schauder alternative theorem). Let be a completely continuous operator (i.e., a map that restricted to any bounded set in is compact). LetThen either the set is unbounded, or has at least one fixed point.
The operator is defined bywhere Accordingly, we know thatand

Lemma 5. The operator is completely continuous.

Proof. The operator is continuous in view of the continuity of , , and Let be bounded. Then there are positive constants and such that For convenience, we setFor any we haveMeanwhile, for , we can getFurthermore, for , we haveHence, the following result can be derivedThus, we have shown the operator is uniformly bounded.
Next, we will show that is equicontinuous. Let with , then we haveandSo we haveHence, we can getwhich implies that as . Therefore, the operator is equicontinuous, and the operator is completely continuous.

3. Main Results

In the following discussion, we assume that the following hypotheses are valid, where and are positive constants, for , (H₁)(H₂) for (H₃)(H₄) for (H₅) for

The first result is based on the Letay–Schauder alternative theorem.

Theorem 1. Assume that (H₁) and (H₂) hold. In addition it is assumed thatwhere Then boundary value problems (4) and (5) have at least one solution.

Proof. It will be verified that the set is bounded. Let then For all we haveAccording to (H₁), (H₂) and Lemma 3, for we haveandAnalogously, we haveaccordingly, we can getHence, we havewhich means that and is bounded. Therefore, by Lemma 4, the operator has at least one fixed point. So boundary value problems (4) and (5) have at least one solution.
Next, we will prove the uniqueness of solutions to boundary value problems (4) and (5) via the Banach contraction mapping principle.

Theorem 2. Suppose that (H₃)–(H₅) are true, in addition thatthen there is a unique solution for boundary value problem (4) and (5).

Proof. For convenience, we denoteWe set , for , on the basis of (H₁) and (H₃), we haveAccording to (H₄) and (H₅), we haveSo for , we haveOn the other hand, we getfurther,In consequence,Therefore, providedwe have
For , we obtainAccording to (H₃)–(H₅), we getSimilarly, it holds thatBased on the above derivation, we conclude thatSo ifthen boundary value problem (4) and (5) has one and only one solution.

Example 1. Consider the following fractional order boundary value problemwith multistripe and band-like boundary conditionswhereClearly,It is easy to verify that (H₁)–(H₅) hold. And by calculation, the following results can be obtained,Furthermore, we haveBy Theorem 1, boundary value problems (63) and (64) have at least one solution. We also haveBy Theorem 2, boundary value problems (63) and (64) have a unique solution.

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 to the manuscript, read, and approved the final draft.

Funding

The work is supported by National Training Program of Innovation. The funding body plays an important role in the design of the study and analysis, calculation and in writing the manuscript.

Acknowledgments

The authors would like to thank the anonymous referees very much for helpful comments and suggestions which led to the improvement of presentation and quality of the work.