The existence of solutions to a class of boundary value problems with fractional difference equations

In this paper, we study the existence and uniqueness of solutions for the boundary value problem of fractional difference equations {-Delta(nu)y(t)=f)t+nu-1,y(t+nu-1)), y(nu-3)=0, Delta y(nu-3)=0, y(nu+b)=g(y) and {-Delta(nu)y(t)=lambda f(t+nu-1,y(t+nu-1)), y(nu-3)=0, Delta y(nu-3)=0, y(nu+b)=g(y) respectively, where t = 1,2,..., b, 2 <nu <= 3, f:{nu-1,...,nu+b}x R -> R is a continuous function and g is an element of subset of([nu-3,nu+b]Z(nu-3), R) is a continuous functional. We prove the existence and uniqueness of a solution to the first problem by the contraction mapping theorem and the Brouwer theorem. Moreover, we present the existence and nonexistence of a solution to the second problem in terms of the parameter lambda by the properties of the Green function and the Guo-Krasnosel'skii theorem. Finally, we present some examples to illustrate the main results.