Abstract

In the present research work, we investigate the existence of a solution for new boundary value problems involving fractional differential equations with -Caputo fractional derivative supplemented with nonlocal multipoint, Riemann–Stieltjes integral and -Riemann–Liouville fractional integral operator of order boundary conditions. Also, we study the existence result for the inclusion case. Our results are based on the modern tools of the fixed-point theory. To illustrate our results, we provide examples.

1. Introduction

Fractional calculus has played a very important role in different areas of research (see [1, 2] and the references cited therein). Consequently, fractional differential equations have grasped the interest of many researchers working in diverse applications [3–6]. Recently, several researchers have tried to propose different types of fractional operators that deal with derivatives and integrals of arbitrary orders and their applications. For instance, Kilbas et al. in [2] introduced the properties of fractional integrals and fractional derivatives concerning another function. Some generalized fractional integral and differential operators and their properties were introduced by Agrawal in [7]. Very recently, Almeida in [8] presented a new type of fractional differentiation operator, the so-called -Caputo fractional operator, and extended work of Caputo [2, 9]. Almeida et al. in [10, 11] investigated the existence and uniqueness of the results of nonlinear fractional differential equations involving a Caputo-type fractional derivative with respect to another function, employing the fixed-point theorem and Picard iteration method. Numerous interesting results concerning the existence, uniqueness, and stability of initial value problems and boundary value problems for fractional differential equations with -Caputo fractional derivatives by applying different types of fixed-point techniques were obtained by Abdo et al. [12, 13], Vivek et al. [14], and Wahash et al. [15]. An important application that is controlled by the theory of -fractional differentiation can be found in [16].

In this paper, we investigate a new boundary value problem of fractional differential equations supplemented with nonlocal multipoint, Riemann–Stieltjes integral fractional boundary conditions involving Riemann–Liouville fractional integral operator of order with respect to function given by the formwhere denotes the -Caputo fractional derivatives of orders and , respectively, is the -Riemann–Liouville fractional integral operator of order . The functions and are continuous, whereas and are Riemann–Stieltjes integral and is a function of bounded variation. is a real constant, and are positive constants.

We also study the corresponding inclusion problem that is given bywhere is a multivalued function, where is the family of all subsets of and the other quantities are the same as defined in problem (1).

Notice that this Riemann–Stieltjes integral fractional boundary conditions arise in manifold applications of computational fluid dynamics, distribution methods, and so forth (for example, see [17, 18]).

This paper is organized as follows. In Section 2, we recall some preliminary results and some related definitions. In Section 3, we discuss the existence results of solutions by relying on Krasnoselskii fixed-point theorem and Leray–Schauder nonlinear alternative. Also, we present an example. Finally, we describe the inclusion case and deduce the existence of solutions by applying Krasnoselskii’s multivalued fixed-point theorem in Section 4.

2. Preliminaries

For the convenience of the reader, we present here some necessary basic definitions, lemmas, and results which are used throughout this paper [2, 8, 10, 12, 19, 20].

Definition 1. Let be a finite interval and and be the set of absolute continuous functions on . Then, we defineendowed with the norm. The convention endowed with the norm is used.

Definition 2. Let , be an integrable function defined on , and be an increasing differentiable function such that for all . The left-sided -Riemann–Liouville fractional integral of order of a function is given by

Definition 3. Let be an integrable function and be an increasing differentiable function such that for all . The left-sided -Riemann–Liouville fractional derivative of order of a function is defined bywhere and denote the integer part of the real number .

Definition 4. Let , and be an increasing differentiable function such that for all . The left-sided -Caputo fractional derivative of order of a function is defined bywhere and for . Furthermore, if and , thenThus, if , one has

Lemma 1. Given a function and , then

Lemma 2. Let and ; then,(i) (ii) (iii) (iv) The existence of solutions of problem (1) relies on the following fixed-point theorems [21, 22].

Theorem 1. (Krasnoselskii’s fixed-point theorem). Let be a closed, convex, bounded, and nonempty subset of a Banach space . Let be operators such that(i)  belong to whenever (ii)  is compact and is a contraction mappingThen, there exist such that .

Theorem 2. (Leray–Schauder fixed-point theorem). Let be a closed and convex subset of a Banach space and be an open subset of with . Suppose that is a continuous, compact (that is, is a relatively compact subset of ) map. Then, either(i)  has a fixed point in or(ii)there are (the boundary of in ) and with For computational convenience, we set the following:

Lemma 3. Let ; then, the linear fractional differential equationhas a solution on given by

Proof. We apply -Riemann–Liouville fractional integral of order to both sides of the linear fractional differential equation:We obtainNext, applying -Riemann–Liouville fractional integral of order to both sides of (14), we obtainwhere are arbitrary constants. By Definition 1, general solution (15) can be written asUsing the boundary condition , we obtain . Thus, (16) takes the formApplying the operator , on equation (17), we obtainUsing the boundary condition , we find thatInserting the value of in (17) yields solution (12). The converse follows by direct computation.

3. Main Results

In this section, we prove the existence of solutions of problem (1). We shall assume that are in the Banach space . Let denote the Banach space of all continuous functions on into endowed with the norm . Here, we define an operator associated with problem (1) by

Therefore, problem (1) has a solution if and only if the operator has a fixed point.

For computational convenience, we introduce the notations

Now, we will state and prove the existence result via Krasnoselskii’s fixed-point theorem.

Theorem 3. Let be continuous functions satisfying the conditions: (H1) for all , every and . (H2) There exist continuous nonnegative functions such that and . Then, problem (1) has at least one solution on .

Proof. For a positive number , consider where , and we split into two operators andwhere, on the bounded set by For any , by using (H2), we have Hence, . Next, we show that is a contraction mapping. Let and , so by using (H1), we have By assumption , we obtain that is a contraction mapping. Since are continuous functions, we have as continuous. Also, is uniformly bounded on as Finally, we prove the compactness of the operator . To show this, we define and and take . Thus, we have independent of . Thus, is equicontinuous. So, is relatively compact on . Hence, by the Arzela–Ascoli theorem, is compact on . Thus, the hypotheses of Theorem 6 are satisfied which leads problem (1) to have at least one solution on . Now, we apply Leray–Schauder nonlinear alternative fixed-point theorem to establish an existence result for problem (1).