On a Nonlocal Multipoint and Integral Boundary Value Problem of Nonlinear Fractional Integrodifferential Equations

The aim of this paper is to give the existence as well as the uniqueness results for a multipoint nonlocal integral boundary value problem of nonlinear sequential fractional integrodifferential equations. First of all, we give some preliminaries and notations that are necessary for the understanding of the manuscript; second of all, we show the existence and uniqueness of the solution by means of the fixed point theory, namely, Banach’s contraction principle and Krasnoselskii’s fixed point theorem. Last, but not least, we give two examples to illustrate the results.


Introduction
In the last few years, fractional differential equations have gained much attention among mathematicians because of the rapid growth and for their applicability in several fields, such as physics, biology, economics, control theory, and engineering; for more details about the theory of fractional differential equations and their applications, we recommend the following articles [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] and the references therein. Furthermore, fractional differential equations with multipoint boundary conditions have provoked a great deal of attention by many authors; a lot of works have been published on this topic; for more details, we give the following references [16][17][18][19][20][21][22].
In a recent paper [23], the existence of solutions for a four-point nonlocal boundary value problem of nonlinear integrodifferential equations of fractional order was proven. In [24], the authors discussed the existence of solutions for fractional differential equations with multipoint boundary conditions. The existence and uniqueness of solutions for multiterm nonlinear fractional integrodifferential equations have been studied in [25]. The existence results for sequential fractional integrodifferential equations with nonlocal multipoint and strip conditions were established in [26], and finally, in [27], the authors studied the existence of solutions for nonlinear fractional integrodifferential equations.
Motivated by all these works, and by the fact that there are no papers dealing with nonlinear fractional integrodifferential equations with multipoint and integral boundary value conditions, in this work, we consider the existence and uniqueness of solutions for the following problem: where This paper is organized as follows: in the second section, we give some preliminaries and notations that will be useful throughout the work; after that, in the third section, we establish the main results by using the fixed point theory; and in the last section, we give some examples to illustrate the results.

Preliminaries and Notations
Throughout this section, we present some notations, definitions, and lemmas which will be used for the rest of the paper.
Definition 1 (see [5]). The fractional integral of order α > 0 with the lower limit zero for a function f can be defined as Definition 2 (see [5]). The 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.
Theorem 3 (see [28]). Let M be a bounded, closed, convex, and nonempty subset of a Banach space X. Let A and B be two operators such that (i) Ax + By ∈ M whenever x, y ∈ M (ii) A is compact and continuous (iii) B is a contraction mapping Then, there exists z ∈ M such that z = Az + Bz.

Journal of Function Spaces
This means that and by using the condition x′ð0Þ = 0, we get θ 0 = 0. As a result of the condition Now, we use the condition μ 1 xð1Þ Finally, we have By substituting the value of θ 0 , θ 1 , and θ 2 , we get the following: Conversely, by direct computations, we obtain the desired result.

Main Results
Let X be the Banach space of all continuous functions from ½0, 1 → ℝ endowed with norm kxk = sup f|xðtÞ| : t ∈ ½0, 1g.
We consider the following set B r = fx ∈ X : kxk ≤ rg, where r ≥ r 2 /ð1 − r 1 Þ, with

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For each t ∈ ½0, 1 and x ∈ B r , we have This means that kPxk ≤ r. Therefore, PB r ⊆ B r . Next, we prove that P is a contraction mapping. For x, y ∈ B r , we have Since r 1 < 1, then P is a contraction. Therefore, system (1) has a unique solution.
Theorem 8. Assume that (H 1 ) holds and f : ½0 ; 1 × ℝ 3 → ℝ is a continuous function. Furthermore, we suppose Then, problem (1) has at least one solution on [0,1] if R < 1, where Proof. We now consider the closed ball B r ′ = fx ∈ X : kxk ≤ r ′ g with fixed radius r ′ : We define the operators P 1 and P 2 on B r ′ as For x, y ∈ B r′ , we have Consequently, 5 Journal of Function Spaces