Abstract

This paper is related to frame a mathematical analysis of impulsive fractional order differential equations (IFODEs) under nonlocal Caputo fractional boundary conditions (NCFBCs). By using fixed point theorems of Schaefer and Banach, we analyze the existence and uniqueness results for the considered problem. Furthermore, we utilize the theory of stability for presenting Hyers-Ulam, generalized Hyers-Ulam, Hyers-Ulam-Rassias, and generalized Hyers-Ulam-Rassias stability results of the proposed scheme. Finally, some applications are offered to demonstrate the concept and results. The whole analysis is carried out by using Caputo fractional derivatives (CFDs).

1. Introduction

It has been observed that the focus of investigation has shifted from classical integer-order models to fractional-order models. It is because of the fact that many practical systems are excellently described by using fractional-order differential equations (FODEs) instead of classical differential equations. For basic theory and some important applications of fractional-order derivatives, we refer the readers to see [1–4] and the references therein. Many researchers are devoted to work in this area and made significant contribution in this regard; we refer the readers to the recent work in [5–10].

The study of implicit systems of FODEs with impulsive conditions is quite important as such systems appear in a variety of problems of applied nature, especially in biosciences, economics, engineering, etc. Such problems arise due to abrupt changes in the state of systems like earth quack, fluctuation of pendulum, etc. Here, we refer to some recent papers on impulsive problems [11–16]. The important class of FODEs known as IFODEs has been given much devotion by researchers. One of the most important aspects is investigation of problems under boundary conditions. Such problems mostly occur in engineering. Boundary and initial conditions may be local or nonlocal and both are important, and increasingly many problems have been investigated under these conditions. Replacing the local conditions by nonlocal ones produces a significant effect. This is due to the fact that the measurement computed from a nonlocal condition is usually more precise than the only one measurement given by a local condition. Therefore, the area of nonlocal boundary value problems has also attracted enough attention. In the last two decades, the area of IFODEs has been investigated from various directions including qualitative theory of existence of solution/solutions, stability, and numerical analysis. Therefore, IFODEs have also been investigated under nonlocal boundary conditions. For instance, Gupta and Dabas [17] studied the existence and uniqueness results for a class of IFODEs with nonlocal boundary conditions.

By employing the fixed point technique, the authors obtained the existence and uniqueness results.

This paper can be considered as generalization of the aforesaid work, in which we discuss existence, uniqueness, and various stability results for the following implicit IFODEs with three point NCFBCs of order :

In the proposed problem, the notation represent Caputo fractional derivatives of orders , and , respectively, where the points in the subscript of the differential operator are actually the limits of the definite integral involved in the definition of CFD. The function is continuous, where is the set of real numbers. The impulsive functions and in C(R, R) are bounded. For the sequence , we have and , and represents the right-hand and left-hand limits of at , respectively, with . The speciality of this proposed problem is that the nonlinear term depends not only on the unknown function but also on its fractional derivative compared with the available results in the literature. This type of study has rarely been discussed in the literature because of the complexity of fractional impulsive surfaces. The further organization of this manuscript is divided into four parts as follows: The second part of the paper demonstrates the preliminary portion in which we recall to readers the basics of used theory, notations, and definitions. The third part presents an existence result by employing Schaefer’s fixed point theorem. The fourth section is introduced to analyze and study several stability results of the considered problem, and the last section is provided to illustrate the applications of the obtained results.

2. Preliminaries

We take , , and . We introduce the following space of piecewise continuous functions bywhere is the Banach space corresponding to the norm .

Definition 1 (see [18]). The fractional order integral of function of order is defined bywhere is the gamma function.

Definition 2 (see [18]). For a function given on interval , the CFD of is defined bywhere .
Let there exist constants and a nondecreasing function , such that the following inequalities exist for :

Definition 3 (see [19]). If for there exists a constant such that for any solution of inequality (6), there is a unique solution of problem (2) which satisfiesthen problem (2) is called Hyers-Ulam stable.

Definition 4 (see [19]). If for and set of positive real numbers there exists , such that for any solution of inequality (6), there is a unique solution of problem (2) which satisfiesthen problem (2) is called generalized Hyers-Ulam stable.

Definition 5 (see [19]). If for there exists a real number , such that for any solution of inequality (8), there is a unique solution of problem (2) which satisfiesthen problem (2) is called Hyers-Ulam-Rassias stable with respect to .

Definition 6 (see [19]). If there exists constant , such that for any solution of inequality (7), there is a unique solution of problem (2) which satisfiesthen problem (2) is called generalized Hyers-Ulam-Rassias stable with respect to .
Here, it is to be noted that Definitions 3–6 have been adopted from the paper [19].

Remark 1. The function is called a solution for inequality (6) if there exists a function together with a sequence , where (which depends on ) such that(i)(ii)(iii)(iv)

Remark 2. A function is a solution of inequality (8) if there exists a function and a sequence , where (which depends on ) such that(i)(ii)(iii)(iv)

Lemma 1 (see [20]). For , the given result holds:

To investigate the nonlinear IFODE2, we first consider the associated linear problem and obtain its solution.

Lemma 2. Let and be continuous. A function is a solution of the fractional integral equation:if and only if is a solution of the following BVP:

Proof. Let for , be the solution of (15), then by Lemma 1, we haveUsing the condition , we getSubstituting in (16), we getFor , we getApplying the impulsive condition , we getSubstituting in (19), we getFrom equations (18) and (22), we getNow, using the impulsive condition , we getSubstituting in (22), we getFor , we getApplying the impulsive condition in (26) and (25), we getSubstituting in (26), we getBy equations (25) and (28), we getNow, using the impulsive condition , we getSubstituting in (28), we getSimilarly for , we getUsing the boundary conditions, , we getBy substituting the value of and summarizing, we get the required result.
Conversely, assume that u satisfies the impulsive fractional integral equation (8); then by direct computation, it can be seen that the solution given by (14) satisfies (15).