Existence and Uniqueness of Solutions for Fractional Integro-Differential Equations Involving the Hadamard Derivatives

In this paper, we study the existence and uniqueness of solutions for the following fractional boundary value problem, consisting of the Hadamard fractional derivative: (H)D(alpha)x(t) = Af (t, x(t)) + Sigma(k)(i=1) C-i (H)I(beta i)g(i) (t, x (t)), t is an element of (1, e), supplemented with fractional Hadamard boundary conditions: (H)D(xi)x(1) = 0, (H)D(xi)x(e) = a D-H(alpha-xi-1/2) ((H)D(xi)x(t))vertical bar(t=delta), delta is an element of (1, e) , where 1 < alpha <= 2, 0 < xi <= 1/2, a is an element of (0, infinity), 1 < alpha - xi < 2, 0 < beta(i) < 1, A, C-i, 1 <= i <= k, are real constants, D-H alpha is the Hadamard fractional derivative of order alpha and I-H(beta i) is the Hadamard fractional integral of order beta(i). By using some fixed point theorems, existence and uniqueness results are obtained. Finally, an example is given for demonstration.