Existence of a positive solution to a system of discrete fractional boundary value problems

We analyze a system of discrete fractional difference equations subject to nonlocal boundary conditions. We consider the system of equations given by -Δ^ν_iy_i(t)=λ_ia_i(t+ν_i-1)f_i(y₁(t+ν₁-1),y₂(t+ν₂-1)), for t∈[0,b]_N0, subject to y_i(ν_i − 2) = ψ_i(y_i) and y_i(ν_i + b) = ?_i(y_i), for i = 1, 2, where ψ_i,?_i:R^b+3→R are given functionals. We also assume that ν_i ∈ (1, 2], for each i. Although we assume that both a_i and f_i(y₁, y₂) are nonnegative for each i, we do not necessarily presume that each ψ_i(y_i) and ?_i(y_i) is nonnegative for each i and each y_i ? 0. This generalizes some recent results both on discrete fractional boundary value problems and on discrete integer-order boundary value problems, and our techniques provide new results in each case.