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

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.


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][2][3][4][5][6][7][8][9][10][11][12][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][15][16]).
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][24][25][26][27][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: (t, x(t), ϕx(t), ψx(t))), t ∈ [0, 1], where λ, δ: 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. e main results in this paper can be viewed as an extension of those provided in [22]. is 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.

Preliminaries and Notations
In this section, we state some notations, definitions, and lemmas which are used in this paper.
Definition 1 (see [5]). e fractional integral of order α > 0 with the lower limit zero for a function f can be defined as Definition 2 (see [5]). e Caputo derivative of order α > 0 with the lower limit zero for a function f can be defined as where n ∈ N, 0 ≤ n − 1 < α < n, t > 0.

A set of functions in C[a, b] is relatively compact if and only it is uniformly bounded and equicontinuous on [a, b].
Theorem 3 (see [30]). If a set is closed and relatively compact, then it is compact.
Lemma 1 (see [5]). Let α, β ≥ 0; then, the following relation holds: Lemma 2 (see [5]). Let n ∈ N and n − 1 < α < n. If f is a continuous function, then we have Lemma 3. Let y ∈ C([0, 1], R). en, a unique solution of the following boundary value problem: is given by Mathematical Problems in Engineering , , Proof. By Lemma 2, we have where Mathematical Problems in Engineering 3 In view of x(0) � λ 1 x(1), we have Substituting the value of c 0 , c 1 , and c 2 , we get the desired results. Directly computing, one can prove the converse of the lemma.

Main Results
Denote by X the Banach space of all continuous functions from ))‖σ‖, and σ will be defined later.
By Lemma 3, we transform problem (1) into a fixed point problem as x � Px, where P: X ⟶ X is given by

Mathematical Problems in Engineering
For computational convenience, we set , Theorem 4. Suppose that en, problem (1) has at least one solution. .
We introduce the decomposition P � P 1 + P 2 , where  For x, y ∈ B r′ , we have

y(s), ϕy(s), ψy(s))ds
6 Mathematical Problems in Engineering us, . (20) Hence, P 1 x + P 2 y ∈ B r′ . For x, y ∈ B r′ , consider By using the condition of the new norm, we have that P 1 is a contraction.
Next, we will show that P 2 is compact and continuous.
Mathematical Problems in Engineering 7 us, P 2 y t 2 − P 2 y t 1 ⟶ 0, as t 1 ⟶ t 2 independently from y ∈ B r′ . (24) is proves that the operator P 2 is relatively compact on B r′ . en, by the Arzelá-Ascoli theorem, we have that P 2 is compact on B r′ . erefore, problem (1) has at least one solution on B r′ . □ Theorem 5. Assume that f: [0, 1] × R 3 ⟶ R is continuous function satisfying en, there exists a unique solution for the boundary value problem (1) provided that Proof. Set sup 0≤t≤1 |f(t, 0, 0, 0)| � M.
en, B r is a closed, convex, and nonempty subset of the Banach space X.
Our objective is to show that the operator P has a unique fixed point on B r .
We prove that PB r ⊆ B r . Mathematical Problems in Engineering which implies that ≤ r 1 r + r 2 ≤ r.

Conclusion
In this paper, new existence and uniqueness results have been studied for nonlinear fractional integro-differential equations equipped with nonseparated conditions. By using a new norm, we have established the new existence of solution for (1) under some weak conditions. e uniqueness of solutions has been given by using the sup norm and applying the Banach fixed point theorem. We have ended the article with some examples to illustrate the results.

Data Availability
No data were used to support this study.