New Existence of Solutions for Fractional Integro-Differential Equations with Nonseparated Boundary Conditions

Abstract

Results reported in this article prove the existence and uniqueness of solutions for a class of nonlinear fractional integro-differential equations supplemented by nonseparated boundary value conditions. We consider a new norm to establish the existence of solution via Krasnoselskii fixed point theorem; however, the uniqueness results are obtained by applying the contraction mapping principle. Some examples are provided to illustrate the results.

1. Introduction

Fractional differential equations have been an important tool to describe many problems and processes in different fields of science. In fact, fractional models are more realistic than the classical models. Fractional differential equations appear in many fields such as physics, economics, image processing, blood flow phenomena, aerodynamics, and so on. For more details about fractional differential equations and their applications, we provide the following references [1–13].

Recently, fractional integro-differential equations were investigated by many researchers in different problems, and a lot of papers were published in this matter (see, for example, [14–16]).

Furthermore, many boundary conditions were considered for the fractional-order integro-differential equations; some of these conditions are the classical, periodic, anti-periodic, nonlocal, multipoint, and the integral boundary conditions (for details, see [17–22]).

On the other hand, many papers have considered the nonseparated boundary conditions as they are a very important class of boundary value conditions (we refer the readers to [23–28]).

Motivated by the above discussion, in this paper, we establish the existence and uniqueness of solutions for a class of fractional integro-differential equations with nonseparated boundary value conditions as follows:where , , are Caputo’s fractional derivatives, is a given continuous function, andwhere , with , .

Our motivation comes from the fact that not many papers have considered the existence and uniqueness results of nonlinear integro-differential equations with nonseparated boundary conditions. On top of that, we show the existence results under some weak conditions. The main results in this paper can be viewed as an extension of those provided in [22].

This paper is divided into five sections. In Section 2, we provide some notations and basic known results. In Section 3, we study the new existence results to problem (1) under some weak conditions, and after that, we show the existence and uniqueness using Banach’s contraction principle. In Section 4, we give two examples to illustrate the results. We end the paper with a conclusion.

2. Preliminaries and Notations

In this section, we state some notations, definitions, and lemmas which are used in this paper.

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

Definition 2 (see [5]). The Caputo derivative of order with the lower limit zero for a function can be defined aswhere , , .

Theorem 1 (see [29]). Let be a bounded, closed, convex, and nonempty subset of a Banach space X. Let and be operators such that(i) whenever .(ii) is compact and continuous.(iii) is a contraction mapping.Then, there exists such that .

Theorem 2 (see [30]) . (Arzelá–Ascoli theorem).
A set of functions in is relatively compact if and only it is uniformly bounded and equicontinuous on .

Theorem 3 (see [30]). If a set is closed and relatively compact, then it is compact.

Lemma 1 (see [5]). Let ; then, the following relation holds:

Lemma 2 (see [5]). Let and . If is a continuous function, then we have

Lemma 3. Let . Then, a unique solution of the following boundary value problem:is given bywhere

Proof. By Lemma 2, we havewhere . Using the condition , we getBy applying , we haveNow we use to getIn view of , we haveSubstituting the value of , , and , we get the desired results. Directly computing, one can prove the converse of the lemma.

3. Main Results

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

By Lemma 3, we transform problem (1) into a fixed point problem as , where is given by

For computational convenience, we set

Theorem 4. Suppose that  For all and , we have with .  with .Then, problem (1) has at least one solution.

Proof. Define withWe introduce the decomposition , whereFor , we haveThus,Hence, .
For , considerBy using the condition of the new norm, we have that is a contraction.
Next, we will show that is compact and continuous.
Continuity of implies that the operator is continuous. Also, is uniformly bounded on asSuppose that . We haveThus,This proves that the operator is relatively compact on . Then, by the Arzelá–Ascoli theorem, we have that is compact on .
Therefore, problem (1) has at least one solution on .

Theorem 5. Assume that is continuous function satisfying  For all and , we have with and .Then, there exists a unique solution for the boundary value problem (1) provided that

Proof. Set .
Let be the ball with radius , where , withThen, is a closed, convex, and nonempty subset of the Banach space .
Our objective is to show that the operator has a unique fixed point on .
We prove that .
For , , we havewhich implies thatNow, for and for ,and thenSince , then the operator is a contraction mapping. Thus, problem (1) has a unique solution.

4. Examples

Example 1. Consider the following fractional problem:We chooseWe haveClearly, , .
Hence, by Theorem 4, problem (31) has a least one solution.

Example 2. Consider the following boundary value problem:Here , , ,