EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A FRACTIONAL BOUNDARY VALUE PROBLEM WITH DIRICHLET BOUNDARY CONDITION

The authors consider a nonlinear fractional boundary value prob- lem with the Dirichlet boundary condition. An associated Green's function is constructed as a series of functions by applying spectral theory. Criteria for the existence and uniqueness of solutions are obtained based on it.

Fractional differential equations have extensive applications in various fields of science and engineering.Many phenomena in viscoelasticity, electrochemistry, control theory, porous media, electromagnetism, and other fields, can be EJQTDE, 2013 No. 55, p. 1 modeled by fractional differential equations.We refer to the reader [10,14] and references therein for discussions of various applications.
The existence of solutions is an essential problem for BVPs involving fractional differential equations.This problem has been studied by many authors, for example, see [1][2][3][5][6][7][8]11,12,15,17] and references therein.As for integer order BVPs, Green's functions play an important role in the study of existence of solutions.However, due to the complexity of the fractional calculus, the Green's functions for fractional BVPs have not yet been well developed.In 2005, Bai and Lü [3] found that is the Green's function for the BVP consisting of the equation and (1.2).This result was obtained by expressing the general solution of the equation −D α 0+ u = h(t) in terms of the α-th Riemann-Liouville integral of h as defined by (1.6) However, this method fails to work for the case when Eq. (1.5) is replaced by a more general equation due to the complexity caused by the extra term a(t)u.Recently, the present authors studied the problems consisting of Eq. (1.7) with a(t) ≡ a, a constant, and the BC u(0) = 0, u(1) = aI α 0+ u(1).By using spectral theory, we derived the Green's function for this problem as a series of functions.By a similar approach, we also obtained the Green's function EJQTDE, 2013 No. 55, p. 2 for the BVP as a series of functions.We refer the reader to [7, Theorem 2.1] and [8, Theorem 2.1] for details.However, we would like to point out that there is a significant restriction in these two problems: the constant a in the second part of the BCs must be the same as the one in the equation.This unnatural assumption is required by technical arguments in the proofs.
In this paper, by applying spectral theory in a different way, we extend the Greens functions in the above problems to the BVP consisting of the equation (1.7) with the Dirichlet BC (1.2).We are then able to obtain results on the existence and uniqueness of solutions of BVP (1.1), (1.2).Our work provides a new approach for constructing Green's functions for fractional BVPs.This method can be further extended to BVPs with more general BCs.
This paper is organized as follows: After this introduction, our main results are stated in Section 2. Two examples are also given there.All the proofs are given in Section 3.

Main results
Throughout this paper, we assume (H) There exists a > 0 such that where G 0 is defined by (1.4) and We then have the following result.With the Green's function G given in Theorem 2.1, we may apply some fixed point theorems to establish criteria for the existence of solutions of BVP (1.1), where a is defined in (H).Then we have the following theorem on the existence of a unique solution.
To illustrate the application of our results, let us consider the following examples.We assume α ∈ (1, 2) and a satisfies (H).EJQTDE, 2013 No. 55, p. 4 Example 1.Consider the BVP where 0 < p < 1/ Then by Theorem 2.2, BVP (2.5) has a unique solution.The solution is nontrivial since Then, the conditions of Theorem 2.3 are satisfied, so BVP (2.6), has at least one nontrivial solution.Note that f does not satisfy the Lipschitz condition in x when x is near 0, and the solution may not be unique.

Proofs
The following lemma on the spectral theory in Banach spaces will be used to prove Theorem 2.1; see [16, page 795, items 57b and 57d] for details.
Lemma 3.1.Let X be a Banach space, A : X → X be a linear operator with the operator norm A and spectral radius r(A) of A. Then: A n , where I stands for the identity operator.
Proof of Theorem 2.1.For any h ∈ X, let u be a solution of the BVP consisting of the equation and BC (1.2).By (1.4), Define A and B : X → X by (Ah)(t) = i.e., (3.7) holds for n = m + 1.By induction, (3.7) holds for any n ∈ N 0 .
Proof of Theorem 2.2.Define T : X → X by Note that K 1 0 G(s)w(s)ds < 1.Hence, T is a contraction mapping.By the contraction mapping principle, T has a unique fixed point.Thus, BVP (1.1), (1.2) has a unique solution.