This paper is concerned with a pair of discrete fractional difference equations
for subject to boundary conditions
where and and are functionals. A standard transformation of the problem is achieved by means of the Green’s function method, and then using the Krasnoselski fixed point theorem for mappings on Banach spaces with cones, two existence theorems are derived for positive solutions of the above boundary value problem based on further restrictions on and the functionals. Since the system is more general than some existing ones, the existence results are extensions of some, if not all, of the existing results for second order difference boundary value problems.