On Antiperiodic Nonlocal Three-Point Boundary Value Problems for Nonlinear Fractional Differential Equations

In recent years, several kinds of boundary value problems of nonlinear fractional differential equations, supplemented with a variety of boundary conditions (including Dirichlet, Neumann, mixed, periodic, antiperiodic, multipoint, integral type, and nonlocal), have been investigated by several researchers. This investigation includes a wide collection of results ranging from theoretical to analytic and numerical methods. For details and examples, see [1–9] and the references therein. This surge has been mainly due to the extensive applications of fractional operators in basic and technical sciences and engineering. One can easily witness from literature (special issues and books) on the topic that the tools of fractional calculus have helped in improving the mathematical modeling of several phenomena of practical nature; for instance, see [10–17]. In this paper, we study a new class of problems of fractional differential equations supplemented with antiperiodic and three-point nonlocal boundary conditions. Precisely, we consider the following fractional problem:


Introduction
In recent years, several kinds of boundary value problems of nonlinear fractional differential equations, supplemented with a variety of boundary conditions (including Dirichlet, Neumann, mixed, periodic, antiperiodic, multipoint, integral type, and nonlocal), have been investigated by several researchers.This investigation includes a wide collection of results ranging from theoretical to analytic and numerical methods.For details and examples, see [1][2][3][4][5][6][7][8][9] and the references therein.This surge has been mainly due to the extensive applications of fractional operators in basic and technical sciences and engineering.One can easily witness from literature (special issues and books) on the topic that the tools of fractional calculus have helped in improving the mathematical modeling of several phenomena of practical nature; for instance, see [10][11][12][13][14][15][16][17].
In this paper, we study a new class of problems of fractional differential equations supplemented with antiperiodic and three-point nonlocal boundary conditions.Precisely, we consider the following fractional problem:

Auxiliary Lemma and Notations
In order to define the solutions for the given problem, we consider the following lemma.
In view of Lemma 1, we consider a fixed point problem equivalent to the nonlinear antisymmetric problem (1) as follows: where the operator H : D → D is defined as Next we set

Main Results
In this section, we present our main results.The first result relies on classical Banach's contraction mapping principle.
Theorem 2. Let  : [0, 1] × R → R be a continuous function satisfying the Lipschitz condition; that is, it exists ℓ > 0 such that Then problem (1) has a unique solution if ℓ < 1, where  is given by (10).
Proof.In the first step, it will be shown that H  ⊂   , where which implies that H  ⊂   , where we have used (10) Since ℓ < 1 by the given assumption, the operator H is a contraction.Thus, by Banach's contraction mapping principle, there exists a unique solution for problem (1).This completes the proof.
The next existence result is based on the following Schaefer's fixed point theorem [18], Th. 4.3.2.Theorem 3. Let  be a Banach space.Assume that  :  →  is completely continuous operator and the set is bounded.Then  has a fixed point in .
which implies that ‖(H)‖ ≤  1 .Further, we find that This implies that H is equicontinuous on [0, 1].Thus, by the Arzelà-Ascoli theorem, the operator H : D → D is completely continuous.Next, we consider the set and show that the set A is bounded.Let  ∈ A, and then  = ]H, 0 < ] < 1.For any  ∈ [0, 1], we have where  will be fixed later.Then, it is enough to show that the operator H :   → D (given by ( 10)) is such that Now we set where  denotes the unit operator.By the nonzero property of Leray-Schauder degree,  1 () =  − H = 0 for at least one  ∈   .In order to justify condition (20), it is assumed that  = H for some  ∈ [0, 1] and for all  ∈ [0, 1] so that which, on taking norm over the interval [0, 1], yields where  is given by (10).In consequence we have Letting  = /(1 − ) + 1, (20) holds.This completes the proof.
The next result is based on Krasnoselskii's fixed point theorem [18], Th. 4.4.1.
Then problem (1) has at least one solution on [0, 1] if Proof.For  ≥ ‖‖, let us define a closed set (ball)   = { ∈ D : ‖‖ ≤ } and introduce the operators H 1 and H 2 on   as For ,  ∈   , it is easy to show that ‖(H where  is given by (10).
Proof.As a first step, we show that the operator H : D → D defined by (10) Clearly, the right-hand side of the above inequality tends to zero independent of  ∈   as  2 →  1 .Thus, by the Arzelà-Ascoli theorem, the operator H is completely continuous.
Let  be a solution for the given problem.Then, for  ∈ (0, 1), as before, we obtain
Notice that the operator H : P → D is continuous and completely continuous.From the choice of P, there is no  ∈ P such that  = H() for some  ∈ (0, 1).Consequently, the conclusion of Lemma 7 applies and hence the operator H has a fixed point  ∈ P which is a solution of problem (1).This completes the proof.