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 -Δνiyi(t)=λiai(t+νi-1)fi(y1(t+ν1-1),y2(t+ν2-1))-Δνiyi(t)=λiai(t+νi-1)fi(y1(t+ν1-1),y2(t+ν2-1)), for t∈[0,b]N0t∈[0,b]N0, subject to yi(νi − 2) = ψi(yi) and yi(νi + b) = ϕi(yi), for i = 1, 2, where ψi,ϕi:Rb+3→Rψi,ϕi:Rb+3→R are given functionals. We also assume that νi ∈ (1, 2], for each i. Although we assume that both ai and fi(y1, y2) are nonnegative for each i, we do not necessarily presume that each ψi(yi) and ϕi(yi) is nonnegative for each i and each yi ⩾ 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.