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 -@D^@ny(t)=f(t+@n-1,y(t+@n-1)), y(@n-2)=g(y), y(@n+b)=0, where f:[@n-1,...,@n+b-1]"N"""@n"""-"""2xR->R is continuous, g:C([@n-2,@n+b]"N"""@n"""-"""2,R) is a given functional, and 1<@n@?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.