Existence, uniqueness and Hyers-Ulam stability of a fractional order iterative two-point boundary value Problems

In this paper, we consider the fractional order iterative two-point boundary value problem D-C(0+)sigma(omega) over bar (t) = f(t, (omega) over bar (t),(omega) over bar ([2])(t)), t is an element of[0,1], (omega) over bar (0) = A, (omega) over bar (1) = B, where (omega) over bar ([2]) (t) = (omega) over bar ((omega) over bar (t)) and 1 < sigma <= 2 and 0 <= A <= B <= 1. The sufficient conditions are derived for the existence and uniqueness of solutions by applying Schauder fixed point theorem and contractionmapping theorem in a Banach space, respectively. Latter, we derived the sufficient conditions for the existence of unique solution by applying Rus's contraction mapping theorem in a metric space, where two metrics are employed. The Hyers-Ulam stability is also established for the problem.