Abstract

This article concerns with the existence and uniqueness theory of solutions for sequential fractional differential system involving Caputo fractional derivatives of order 1<alpha, beta<2 with coupled nonseparated boundary conditions. The standard tools of the fixed point theory were used to establish the main results. Application is introduced to show the validity of our results.

1. Introduction

This article is devoted to the study of the following nonlinear sequential fractional differential equation.where is the fractional derivative of order in Caputo sense, and , and

The studied equation, equation (1), is subject to boundary conditions which are coupled but nonseparated. The main goal of the article is to prove the existence of solutions to the problem defined by equation (1).

The theory of fractional differential equation subject to boundary conditions has long been of the interest of researchers. Two-point boundary conditions are common in bibliography; however, special attention is given to more complex boundary conditions such as integral and nonlocal multiple points.

Differential equations with nonlocal boundary conditions are of high importance in applied sciences. They are useful in modeling some chemical processes and in general to model a process which is located inside a defined region. On the other hand, differential equations with integral boundary conditions are appropriate for modeling problems that involve flowing, such as blood flow problem (see [1, 2]). Other application examples are biology, finances, etc. (see [18–20] for illustration). Some interesting general results on boundary value problems are found in [3, 10–17] and [21–23]. In references [4–9, 24–26], the authors have investigated sequential fractional differential equations.

Although many articles have focused on the study of fractional differential equations with multiple boundary conditions, there is no or there are few works in the literature that studied problems with coupled nonseparated fractional boundary conditions. In this regard, we study the problem, equation (1), subject to nonseparated fractional boundary conditions.

To prove an existence and uniqueness of a solution to problem defined by equation (1), two main theorems are proposed in the work. Banach’s fixed point theorem is used to establish the first theorem in which uniqueness of a solution to problem (1) is proved. The second theorem establishes existence of the solution to problem (1), using Leray–Schauder’s alternative criterion. After this introductory section of this work, there are four other sections, which are organized following a hierarchical structure. The second section provides a brief review of fractional calculus definitions. The third section is dedicated to the main results of the work. The fourth section provides a numerical example to back up the theoretical results. Finally, conclusion and possible future works are discussed in the fifth section.

Notations:wherewith

Then,

In a like manner,

Lemma 1. Let , then the solution of the following system:is given by

Proof. Solving the sequential linear equationswe getNow, we need to find the constants and . Before finding these constants, we find,The first boundary conditions give,From the second boundary conditions , we get,A (15) and (17) implies,Next, we substitute (18) and (19) into (14) and (16) and solve the result system of the equations; this leads to,By finding the constants and and substituting them in (12), we get the solution, proof completed.