Existence and Stability Results for Caputo-Type Sequential Fractional Differential Equations with New Kind of Boundary Conditions

Abstract

In this paper, we present the existence and the stability results for a nonlinear coupled system of sequential fractional differential equations supplemented with a new kind of coupled boundary conditions. Existence and uniqueness results are established by using Schaefer’s fixed point theorem and Banach’s contraction mapping principle. We examine the stability of the solutions involved in the Hyers–Ulam type. A few examples are presented to illustrate the main results.

1. Introduction

In recent years, it has been demonstrated that the definition of fractional calculus is more suitable for describing historical dependence processes than the local limit definitions of ordinary or partial differential equations, and it has garnered increasing attention in a variety of fields [1–4]. Fractional calculus have arisen as a significant topic of research due to the multiple applications of their techniques in scientific and technical disciplines, such as monetary economics, ecological economics, aerodynamics, marketing, mathematical physics, mathematical biology, aeronautics, and financial mathematics [5–11]. Numerous scholars have become interested in this area of mathematical analysis as a result of its popularity and have contributed to its numerous facets. The boundary value problems of fractional order, in particular, garnered much attention. See [12–19] for the most recent results on FDEs with multipoint and integral boundary conditions.

Numerous authors have also examined coupled systems of differential equations of fractional order. These systems naturally exist in a wide variety of real-world circumstances [19–22]. A series of papers [19, 20, 23–27] and the sources listed therein contain some recent results on this subject. More recently, in [28], the authors discussed the existence and uniqueness of coupled system of nonlinear FDEs with a new kind of coupled boundary conditions specified bywhere is the Caputo fractional derivatives (CFDs) of order , , are continuous functions, and is a non-negative constant. The authors proved the existence and uniqueness results with the aid of standard fixed point to obtain their results. Recently, Subramanian et al. [29] studied the existence of positive solutions for nonlinear coupled system of fractional differential equations complemented with boundary conditions.

In [19], authors studied the existence for a the system of nonlinear coupled Caputo-type SFDEs and inclusions subject to multipoint and fractional integral boundary conditions of the formandwhere , , are continuous functions, is the collection of nonempty subsets of , and is positive constant. The author investigation is mainly based on the theorems of Schaefer’s, Banach’s, Covitz–Nadler, and nonlinear alternatives for kakutani.

In this article, we consider the following system of nonlinear sequential fractional differential equations,where and are the Caputo fractional derivatives (CFDs) of orders , are continuous functions, and is a non-negative constant. The boundary conditions (BCs) introduced in the problem (4) can be interpreted as the sum of the unknown functions and at the interval equals zero, while the contribution of sum of the unknown functions on an arbitrary domain of the given interval remains constant. Existence and uniqueness of the above mentioned nonlinear system is investigated. Unlike in [28], the main results of this article are entirely different. Because we consider the problems in the context of sequential fractional differential equations, unique techniques based on Schaefer’s and Banach’s contraction mapping principle are employed. In addition, the Ulam–Hyers stability technique for the problem (4) is also investigated, while it was not considered in [28, 30–33].

The rest of the paper is organized as follows: in Section 2, we recall some basic definitions from fractional calculus and present an auxiliary result, which plays a pivotal role in transforming system into equivalent integral equations. In Section 3, the existence results for the problem at hand are proved via the standard fixed point theorems. In Section 4, we present certain criteria under which the proposed problem is Ulam-Hyers stable. Furthermore, as an application, two examples are given to shows the applicability of the obtained results.

2. Preliminaries

In this section we present some definitions and auxiliary results that will be used to prove our main theorems (see [13, 14, 21, 22, 33, 34]).

The space of Lebesgue measurable functions , where , and

Definition 1. The generalized Riemann–Liouville fractional integral (GRLFI) of order and , of a function , for all , is defined asandfor are called the left and right sided generalized Riemann–Liouville fractional integral (GRLFI) of order , respectively. The operators and are defined for .

Remark 1. The above definition for generalized Riemann–Liouville fractional integral (GRLFI)s reduce to the RLFIs for .and

Definition 2. The RL fractional derivative of order , , , is defined aswhere the function has absolutely continuous derivative up to order .

Definition 3. The Caputo fractional derivative (CFD) of order for a function can be written as

Remark 2. If , then,The following auxiliary lemma, which concerns the linear variant of problem (4) plays a key role in the sequel.

Lemma 1. Let and . The solution of the linear system of FDEsis given by

Proof. Solving,Using the conditions under which in (17), we find that and and using the boundary conditions , and in (17), we obtainSubstituting the values of and in (17) leads to the solution (14) and (15). The proof is complete.

3. Existence Results

Denote the Banach space by endowed with the norm , for . In view of Lemma 1, we introduce the operator associated with the problem (4) as followswhere

Next, the following assumptions will be used to demonstrate the paper’s results.

Let be continuous functions.

There a exist continuous non-negative function such that

There exist non-negative constants , and such that .

To facilitate the computations, we introduce the notation:and

In this part, we prove the existence of a solution to the BVPs via fixed point theorem of Schaefer’s.

Theorem 1. Assume that holds. In addition, the assumption is thatwhere are defined by (24) and (25). Then, the problem has at least one solution on .

Proof. In the first part, we demonstrate that the operator is completely continuous. The continuity of the operator comes from the continuity of the function of and . Following that, we show that the operator is continuously bounded. Now let be bounded. Then, there exist non-negative and constants such thatSo, for any , we getThus,Hence, it follows from the above inequality that the operator is uniformly bounded.
In order to show that maps bounded sets into equicontinuous sets of , let , , and . Then,similarly,Note that the right-hand sides of the above inequalities tend to zero as and are independent of . Thus, it follows by the Arzela–Ascoli theorem that the operator is completely continuous. Next, we consider the set . For any , we haveUsing given by (24) and (25), we find that