Abstract: In this paper, we investigate the nonlinear fractional difference equation with nonlocal fractional boundary conditions. We derive the Green’s function for this problem and show that it satisfies certain properties. Some existence results are obtained by means of nonlinear alternative of Leray-Schauder type theorem and Krasnosel-skii’s fixed point theorem. 

Key words: discrete fractional calculus, green's function, nonlocal fractional boundary conditions, existence of solution, fixed point