Existence and Uniqueness Results for a Fractional Differential Equations with Nonlocal Boundary Conditions

abstract: In this paper, we consider a boundary value problem of differential equations of fractional order involving the nonlocal boundary condition. We establish sufficient conditions for the existence of solution of the boundary value problem with the help of Schaefer’s fixed point theorem. Our uniqueness result is based on contraction mapping principle. As an application, we give two examples that illustrate our results.


Introduction
Fractional differential equations are being used in various fields of science and engineering such as control system, electrochemistry, electromagnetics, viscoelasticity, physics, biophysics, porous media, blood flow phenomena, electrical circuits, biology, fitting of experimental data etc. Due to these features, models of fractional order become more practical and realistic than the models of integer-order. The existence and uniqueness of boundary value problem for fractional differential equations have attracted attention of many authors, see ( [1]- [9]). For some recent development on the topic, see [10,11,12,13], and the references therein. In papers [14,15], the authors consider the stability of fractional differential equations. Besides these cited works, few more contributions [16,17], have been made to the analytical and numerical study of the solutions of fractional integral equations via fixed point theorems.
In [18], Cabrera et al. study the existence and uniqueness of positive solutions to the following nonlinear fourth-order boundary value problem which describes the deflection of an elastic beam with the left extreme fixed and the right extreme is attached to a bearing device given by a known function.
Motivated by the problem in [18], we study the existence and uniqueness of solutions for the following nonlinear fractional boundary value problem with nonlocal boundary condition c D α z(ξ) = w(ξ, z(ξ)), ξ ∈ [0, 1],  In boundary value problem (1.2), the authors consider fractional order derivative but in BVP (1.1) cabrera et al. consider the ordinary derivative of fourth order. So derivative in BVP (1.1) becomes a particular case of derivative in BVP (1.2) for α = 4. Also we consider the non local boundary conditions. As remarked by Byszewski [19,20], the nonlocal condition can be more useful than the standard initial condition to describe some physical phenomena.

Preliminaries
Let us recall some basic definitions and results of fractional calculus.
denotes the integer part of the real number q.
provided that such integral exists.
for some c i ∈ R, i = 0, 1, 2, ..., n − 1, where n is the smallest integer greater than or equal to q.

Auxiliary Result
Here we establish supporting result for the main results of the next section.
is given by Proof. In view of Lemma 2.1, (3.1) is equivalent to and Observe that the fixed point of W are the solution of (1.2).

Main results
In this section, we develop two different types of results for existence and uniqueness of the proposed nonlinear fractional differential equation (1.2) by using Banach contraction principle and Scheafer's fixed point theorem.
First result is based on Banach contraction principle.
Proof. We shall prove W is a contraction.

Proof. We shall prove this result by Schaefer's fixed point theorem
Step I. W is Continuous. Let {z n } be a sequence in Z such that z n → z. Then for each ξ ∈ [0, 1] 1] |w(s, z n (s)) − w(s, z(s))|ds Since w and g are continuous functions, therefore W is also continuous.
Step II. Bounded sets of Z are mapped into bounded sets of Z under the mapping W . Now, for z ∈ B ǫ and ∀ξ ∈ [0, 1], Thus, Step III. W (B ǫ ) is equi-continuous Let z ∈ B ǫ and ξ 1 , ξ 2 ∈ [0, 1] with ξ 1 < ξ 2 , then Now the right-hand side approaches to zero when ξ 1 approaches to ξ 2 . Combining Steps I to III and by the consequence of Arzelá-Ascoli theorem, W is completely continuous operator.
We will show that the set Θ is bounded.
|K(ξ, s)| × |w(s, z(s))|ds Thus, ||z|| < ∞ which implies that Θ is a bounded set. By Schaefer's fixed point theorem, W must have at least one fixed point which is a solution of (1.2). ✷

Examples
In this section, we discuss some examples to illustrate our results.