Abstract

In this study, we investigate the existence of a solution to the boundary value problem (BVP) of variable-order Caputo-type fractional differential equation by converting it into an equivalent standard Caputo (BVP) of the fractional constant order with the help of the generalized intervals and the piecewise constant functions. All our results in this study are proved by using Darbo’s fixed-point theorem and the Ulam–Hyers (UH) stability definition. A numerical example is given at the end to support and validate the potentiality of our obtained results.

1. Introduction

Fractional differential equations of a constant order or fractional calculus, in general, have been studied by many researchers for more than three centuries compared to integer differential equations. In recent years, the notion of a variable-order operator is a much more recent improvement. Different authors have presented different definitions of variable-order differential; we refer to [1–6].

Several investigators have studied boundary value problems (BVPs) for different types of fractional differential equations (FDEs), for example, Adiguzel et al. [7] obtain a solution for a nonlinear (FDEs) of order , Benchora and Souid [8] obtain a solution for implicit fractional-order differential equations, and Zhang [9] discuss the existence of solutions for two point (BVPs) with singular (FDEs) of variable order.

Bai and Kong [10] studied the following problem:where and are the Caputo–Hadamard derivative and Hadamard integral operators of constant order , respectively, is a given continuous function, and .

Some existence and Ulam stability properties for FDEs are studied by many authors (see [11, 12] and references cited therein).

Motivated by the above studies, we deal with the existence of solutions and the stability of the obtained solution in the sense of Ulam–Hyers (UH) to the following BVP of Caputo variable-order type:where , is the variable order of the fractional derivatives, is a continuous function, and the left Riemann–Liouville fractional integral (RLFI) of variable-order for function is (see, for example, [13–15])and the left Caputo fractional derivative (CFD) of variable-order for function is (see, for example, [13–15])

2. Preliminaries

This section introduces some important fundamental definitions and results that will be needed in this paper.

Denote by the Banach space of continuous functions , with the norm

Remark 1. In (2), the variable order , but the RLFI could be defined for any .

Remark 2. In the case of a constant order in equations (2) and (3), the RLFI and CFD coincide with the standard Riemann–Liouville fractional integral and Caputo fractional derivative, respectively (see [13, 14, 16]).
Recall the following properties of fractional derivatives and integrals [16].

Lemma 1. Assume that , , , and . Then, the differential equation,has a unique solution:where and , .

Lemma 2. Let , , , and . Then,where and , .

Lemma 3. Let , , and . Then,

Lemma 4. Let , , and . Then,

Remark 3. It is to be hereby note that the semigroup property is not satisfied for general functions and (see [9, 17, 18]), that is,

Example 1. Assume thatandSo, we obtainTherefore, we obtain

Definition 1 (see [19–21]). Let , where is named a generalized interval if it is either an interval, or or .
A finite set is named a partition of if each in lies in just one among the generalized intervals in .
A function is defined to be piecewise constant with respect to partition of if admits constant values on , for any .
Zhang et al. [22] gave very interesting result.

Lemma 5. If , then both of the following holds:(a)For , (b)For , the variable-order fractional integral exists for any points on

Definition 2 (see [23]). Let be a bounded subset of the Banach space . The Kuratowski measure of noncompactness (KMNC) is a mapping which is defined as follows:whereThe KMNC satisfies the following properties.

Proposition 1 (see [23, 24]). Let be a Banach space and , and be bounded subsets of . Then,(1) if and only if is compact(2)(3)(4) implies (5)(6)(7)(8)(9) for any

Lemma 6 (see [25]). Let be a bounded and equicontinuous set; then,(i)The function is continuous for , and(ii) where

Theorem 1 (DFPT, see [23]). Let be a Banach space and be a nonempty, bounded, closed, and convex subset of and is a continuous operator satisfying.

Then, has at least one fixed point in .

Definition 3 (see [11]). The BVP (2) is (UH) stable if satisfies the following inequality:where solution of BVP (2) with

3. Existence of Solution

In this section, we investigate the existence of solution for a BVP of a Caputo-type fractional differential equation using DFPT and KMNC.

Let us introduce the following assumptions.

(H1) Let be an integer,a partition of the interval , and let be a piecewise constant function with respect to , i.e.,where are constants, and is the indicator of the interval (with ), such that

Remark 4. According to remark of Benchohra (p.20 in [26]), it is not difficult to show that condition (H2) and the following inequality,are equivalent for any bounded sets and for each .
Furthermore, for a given set of functions , we denote flushleft:andThe symbol , which indicated the Banach space of continuous functions equipped with the normwhere ,
Then, for , the left Caputo fractional derivative (CFD), defined by (4), could be presented as a sum of left Caputo fractional derivatives of constant orders ,Thus, by (30), BVP (2) can be written, for any as the following form:Now, we will present the definition of the solution to BVP (2).

Definition 4. BVP (2) has a solution if there are functions , so that fulfilling equation (31) and .
Let the function be a solution of integral (31), such that on . Then, (31) is reduced toWe consider the following auxiliary BVP:The following lemma is necessary in our next analysis of BVP (33).

Lemma 7. Let . Suppose that there exists a number such that , for any .

Then, the solution of BVP (33) can be expressed by the integral equation:

Proof. Let is a solution of BVP (33). Taking (RLFI) to both sides of (33) and using Lemma 2, we findBy , we get .
By , we observe thatThen,Conversely, let be solution of integral equation (34). Employing the operator (CFD) to both sides of (34) and Lemma 3, we deduce that is the solution of BVP (33).
Based on concept of MNCK and DFPT, we have the following theorem for the existence of a solution for BVP (33).