Existence and Uniqueness for the Boundary Value Problems of Nonlinear Fractional Differential Equation

This paper studies the existence and uniqueness of solutions for a class of boundary value problems of nonlinear fractional order differential equations involving the Caputo fractional derivative by employing the Banach’s contraction principle and the Schauder’s fixed point theorem. In addition, an example is given to demonstrate the application of our main results.

Recently some basic theory for the initial value problems of fractional differential equations involving Riemann-Liouville differential operator ( 0 1 α < < ) has been discussed by Lakshmikantham et al. [24] [25] [26].In a series of papers (see [6] [11]), the authors considered some classes of boundary value problems for differential equations involving Riemann-Liouville and Caputo fractional derivatives of order 0 1 α < < and 2 3 α < < .This paper generalizes the results of the papers above [6] and presents some existence theorems for the boundary value problems (BVP) (1.1).Two theorems are based on the Banach fixed point theorem, and the others are based on Schauder's fixed point theorem and Leray-Schauder type nonlinear alternative.An example is given to demonstrate the application of our main results.

Preliminaries
Some notions and Lemmas are important in order to state our results.Denote by ( ) , C J R the Banach space of all continuous functions from J into R with the norm The fractional order integral of the function where Γ is the gamma function.[11]) For a function h given on the interval [a,b], the α - th Caputo fractional-order derivative of h is defined by

Definition 2.2 ([6]
where A solution of the problem (1.1) is defined as follows.

a a a a D I h t h t I I h t I h t
As a consequence of Lemmas 2.1, Lemmas 2.2 and Lemmas 2.3, the following result is useful in what follows.
Lemma 2.4 Let if and only if ( ) x t is a solution of the fractional integral equation And the following simple calculation can be obtained by (2.4) Hence Equation (2.5).Conversely, it is clear that if ( ) x t satisfies Equation (2.5), then Equations (2.4) hold.

Existence and Uniqueness of Solutions
In this section, Our first result is based on the Banach fixed point theorem (see [28]).
Theorem 3.1 Assume that (H1) There exists a function ( ) ( ) Then the BVP (1.1) has a unique solution on J.
Proof.Transform the problem (1.1) into a fixed point problem.Consider the operator ( ) ( ) The Banach contraction principle is used to prove that T has afixed point.Let Consequently, by (3.1) T is a contraction operator.As a consequence of the Banach Fixed point theorem, T has a fixed point which is the unique solution of the problem (1.1).The proof is completed.
In Theorem 3.1, if the function ( ) is replaced by a constant L > 0, the second result follows.Theorem 3.2 Assume that (H2) There exists a constant L > 0 (i.e. ( ) Then the BVP (1.1) has a unique solution on J. has a fixed point.The proof will be given in several steps.
Step 1: T is continuous. ) Step 2: T maps the bounded sets into the bounded sets in ( ) For any * 0 η > , it can be shown that there exists a positive constant  such that In fact, t J ∀ ∈ , by (3.2) and (H4) Step 3: Tmaps the bounded sets into the equicontinuous sets of ( ) be abounded set of ( ) , C J R as above, and As 1 2 t t → , the right-hand side of the aboveinequality tends to zero.As a con- sequence of Steps 1 to 3 together with the Arzelá-Ascoli theorem, ( ) ( ) Step 4: A priori bounds.
) ( ) ( ) ( ) This shows that the set ε is bounded.As a consequence of Schauder's fixed point theorem, T has a fixed point which is a solution of the problem (1.1).
In Theorem 3.3, if the condition (H4) is weakened, the fourth result can be obtained, which is a more general existence result (see [6]).Theorem 3.4 Assume that (H3) and the following conditionshold.
(H5) There exist a functional ( ) and a continuous and nondecreasing , for each , and .
Then the BVP (1.1) has at least one solution on J.  for some ( ) 0,1 λ ∈ .Therefore, T is Leray-Schauder type operator (see [6]), so that it has a fixed point ( ) x t in U , which is a solution of the BVP (1.1).