On the Generalization of a Solution for a Class of Integro-Differential Equations with Nonseparated Integral Boundary Conditions

In this paper, the existence and uniqueness results of the generalization nonlinear fractional integro-differential equations with nonseparated type integral boundary conditions are investigated. A natural formula of solutions is derived and some new existence and uniqueness results are obtained under some conditions for this class of problems by using standard fixed point theorems and Leray–Schauder degree theory, which extend and supplement some known results. Some examples are discussed for the illustration of the main work.


Introduction
Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. Characteristics of the fractional derivatives make the fractional-order models more realistic and practical than the classical integral-order models. Fractional differential equations have gained considerable importance due to their application in various sciences, such as physics, polymer rheology, aerodynamics, capacitor theory, chemistry, biology, control theory, and electrodynamics of complex medium. e initial and boundary value problems for nonlinear fractional differential equations arise from the study of models of viscoelasticity, electrochemistry, porous media, and electromagnetics. In consequence, the subject of fractional differential equations is gaining much importance and attention [1][2][3][4]. e recent development in the theory and methods for fractional differential equations indicates its popularity. For more details, we refer the reader to [5][6][7][8][9] and the references cited therein.
In [20], Ahmad and Nieto investigated the fractional differential equations with antiperiodic fractional boundary conditions as the following form: t denote the Caputo fractional derivative of order α, β; f: [0, T] × R ⟶ R is a given continuous function; and T is a fixed positive constant. e results are based on some standard fixed point principles.
Recently, in [24], the author discussed the nonlinear fractional differential equations with nonseparated type integral boundary conditions where C D α t denotes the Caputo fractional derivative of order α, f, g, h: [0, T] × R ⟶ R are given continuous function, and λ 1 , λ 2 , μ 1 , μ 2 are suitably chosen real constants with λ 1 ≠ − 1, λ 2 ≠ − 1. By applying the Leray-Schauder degree theory and some standard fixed point theorems, some new existence and uniqueness results are obtained.
Motivated by the abovementioned papers and many known results, in this paper, we concentrate on the existence and uniqueness of solutions for the nonlinear fractional integro-differential equations and inclusions of order α ∈ (1, 2], with nonseparated type integral boundary conditions where C D α t and C D β t denote the Caputo fractional derivative of order α, β; f: [0, T] × R × R × R ⟶ R is a given continuous function satisfying some assumptions that will be specified later; Γ is the Euler gamma function; and μ 1 ≠ − 1, denotes the Banach space of all continuous functions from [0, T] to R endowed with a topology of uniform convergence with the norm ‖u‖ � sup |u(t)|, { t ∈ [0,T]}. To the best of our knowledge, no paper has considered the generalization of nonlinear fractional integro-differential equations with nonseparated type integral boundary conditions (3). Our purpose here is to give some existence and uniqueness results for solution to (3).
Compared with the previous research problems, (3) has more general integral boundary value conditions. is paper is organized as follows: in Section 2, we present the notations and give some preliminary results via a sequence of definitions and lemmas. In Section 3, we prove new existence and uniqueness results for problem (3). ese results are based on fixed point theorems and Leray-Schauder degree theory. In Section 4, two examples are demonstrated which support the theoretical analysis.

Preliminaries and Lemmas
In this section, we present some basic notations, definitions, and preliminary results which will be used throughout this paper. Let us recall some definitions of fractional calculus. For more details, see [1,2].
e fractional integral of order α with the lower limit zero for a function f: [0, ∞) ⟶ R is defined as provided the integral exists.

Definition 2.
For a function f: [0, ∞) ⟶ R with the lower limit zero, the Caputo derivative of fractional order α is defined as where n � [α] + 1 and [α] denote the integer part of the real number α.
e Riemann-Liouville fractional derivative of order α with the lower limit zero for a function f(t) is defined by where n � [α] + 1 and [α] denote the integer part of real number α, provided that the right side is pointwise defined on (0, ∞).

Lemma 1. For α > 0, the general solution of the fractional differential equation
where In view of Lemma 1, it follows that In the following, we derive a natural formula of solution to the integral boundary value problem for integro-differential equation (3).
if and only if u is a solution of the integral equation Proof. Assume that y satisfies (10). Using Lemma 1, for some constants c 0 , c 1 ∈ R, we have Applying the boundary conditions for problem (3), we find that Mathematical Problems in Engineering Substituting the value of c 0 and c 1 in (12), we obtain the unique solution of (10) which is given by Conversely, we assume that u is a solution of the integral equation (11), and in view of the relations C D Moreover, it can easily be verified that the boundary conditions are satisfied. e proof is completed. By Lemma 2, problem (3) is reduced to the fixed point problem where Φ: C ⟶ C is given by □

Main Results
In this section, we will show the existence and uniqueness of solutions for problem (3). Now we state some known fixed point theorems which are needed to prove the existence of solutions for equation (3).
Then Φ has a fixed point in Ω.

Mathematical Problems in Engineering
Now, for u, v ∈ C and for each t ∈ [0, T], we obtain Observe that r 1 depends only on the parameters involved in the problem. As r 1 < 1, then Φ is a contraction map. Hence, the conclusion of the theorem follows by the contraction mapping principle, and Φ has a unique fixed point u. at is, the boundary value problem (3) has a unique solution. is completes the proof.
Our next existence results are based on Krasnoselskii's fixed point theorem [40].

Theorem 4. Let M be a closed convex and nonempty subset of a Banach space X. Let A and B be the operators such that (i) Ax + By ∈ M, whenever x, y ∈ M (ii) A is compact and continuous (iii) B is a contraction mapping
en, there exists z ∈ M such that z � Az + Bz.

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For u, v ∈ B R , we find that us, Φ 1 u + Φ 2 v ∈ B R . It follows from the assumptions (H 1 ) and (H 2 ) that Φ 2 is a contraction mapping if Moreover, the continuity of f implies that the operator Φ 1 is continuous. Also, Φ 1 is uniformly bounded on B R as Now, we prove compactness of the operator Φ 1 . In view of (H 3 ), we define and consequently, we have which is independent of u and tends to zero as t 2 − t 1 ⟶ 0. So Φ 1 is relatively compact on B R . Hence, by the Arzelá-Ascoli theorem, Φ 1 is compact on B R . us, all the assumptions of eorem 4 are satisfied. erefore, the conclusion of eorem 4 applies that the fractional boundary value problem (3) has at least one solution on [0, T]. is completes the proof.

(49)
us, all the assumptions of eorem 3 hold. Consequently, the conclusion of eorem 3 implies that problem (46) has a unique solution.

Data Availability
No data were used in the manuscript.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.