The existence and uniqueness of solutions to boundary value problems of fractional difference equations

In this paper, we study the existence and uniqueness of solutions for the boundary value problem -Delta(nu)(nu-mu)y(t) = f (t, y(t+ v-1)), Delta(l)y(nu-N) = 0, i is an element of{0, ... , N-3}, Delta(N-2)y(nu-N) = g(y), Delta(mu)(nu-N)y(b+M+v-mu) = 0, where v >= 2,1 <= mu<nu, f : {0, ..., b+M} x R -> R is continuous, and nonnegative for y >= 0,g : C([v-N, ... ,b+M+v],R) is a given function. We give a representation for the solution to this problem, and we prove the existence and uniqueness of solution to this problem by contraction mapping theorem and Brouwer theorem.