Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions

In this paper, we consider a discrete fractional boundary value problem of the form -Delta(nu)y(t) = f (t + nu - 1, y(t + nu - 1)), y(nu - 2) = g(y), y(nu + b) = 0, where f : [nu - 1, ..., nu+b-1]N nu-2 x R -> R is continuous, g : C([nu - 2, v+b)N nu-2 is a given functional, and 1 < v <= 2. We give a representation for the solution to this problem. Finally, we prove the existence and uniqueness of solution to this problem by using a variety of tools from nonlinear functional analysis including the contraction mapping theorem, the Brouwer theorem, and the Krasnosel'skii theorem. (C) 2010 Elsevier Ltd. All rights reserved.