Abstract

This paper discusses a boundary value problem of nonlinear fractional integrodifferential equations of order and and boundary conditions of the form . Some new existence and uniqueness results are proposed by using the fixed point theory. In particular, we make use of the Banach contraction mapping principle and Krasnoselskii’s fixed point theorem under some weak conditions. Moreover, two illustrative examples are studied to support the results.

1. Introduction

Fractional differential equations are relevant in many fields of science, such as chemistry, fluid systems, and electromagnetic; for more details about the theory of fractional differential equations and their applications, we invite the readers to see [1–16] and the references therein.

Some physical applications of fractional differential equations include viscoelasticity, Schrodinger equation, fractional diffusion equation, and fractional relaxation equation; for more details, we refer the readers to [17].

In addition, fractional integrodifferential equations are used as an important tool to gain insight into some emerging problems from several science areas, for more details, we give the following references [18–23].

More recently, in [24], the existence and uniqueness of positive solutions for the fractional integrodifferential equation were proved.

In [25], the authors discussed the existence and uniqueness of solutions for nonlinear integrodifferential equations of fractional order with three-point nonlocal fractional boundary conditions. The existence of solutions for nonlinear fractional integrodifferential equations has been studied in [26].

Motivated by all these works and by the fact that there are no papers dealing with the new existence results for nonlinear fractional integrodifferential equations, in this work, we consider the existence and uniqueness of solutions for the following problem: where are the Caputo fractional derivatives, is a continuous function, and

where , with , .

This paper is organized as follows. In Section 2, we present some preliminaries and notations that will be required for the later sections. After that, in Section 3, we establish the main results by using the fixed point theory. And, in the last section, we give two examples to illustrate the results.

2. Preliminaries and Notations

In this section, we give some notations, definitions, and lemmas which will be required for the rest of the paper.

Definition 1 [5]. The fractional integral of order with the lower limit zero for a function can be defined as

Definition 2 [5]. The Caputo derivative of order with the lower limit zero for a function can be defined as where , , .

Theorem 3 [27]. Let be a bounded, closed, convex, and nonempty subset of a Banach space . Let and be two operators such that (i) whenever (ii) is compact and continuous(iii) is a contraction mappingThen, there exists such that .

Lemma 4 [5]. Let ; then, the following relation hold:

Lemma 5 [5]. Let and . If is a continuous function, then we have

Lemma 6. Let . Then, the unique solution of the problem is given by

Proof. By applying Lemma 5, we have where . So And by using , we obtain . As a result of , we have that Now, we use to get By substituting the value of , we obtain the following Conversely, by direct computations, we obtain the desired result.

3. Main Results

Let be the Banach space of all continuous functions from endowed with the norms and , where , and will be defined later.

Theorem 7. Assume that
(H₁) for all and , we have with
(H₂) , with . Then, the problem (1) has at least one solution.

Proof. We consider the ball with .We define the operators on , where. For , we have Therefore, Then, Now, we prove that is a contraction. For , we have By using the condition of the new norm, we conclude that is a contraction.
Next, we will prove that is compact and continuous.
Continuity of implies that the operator is continuous. Also, is uniformly bounded on as Suppose that . We have Then, , as independently from .
This shows that the operator is relatively compact on . Thus, by the Arzela Ascoli theorem, we obtain that is compact on .
By the Krasnoselskii fixed point theorem, the problem (1) has at least one solution on .

Theorem 8. Suppose that is a continuous function satisfying
for all and , we have with .Then, there exists a unique solution for the problem (1) under the following condition: , where
with .