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

Abstract We analyze a system of discrete fractional difference equations subject to nonlocal boundary conditions. We consider the system of equations given by - Δ ν i y i ( t ) = λ i a i ( t + ν i - 1 ) f i ( y 1 ( t + ν 1 - 1 ) , y 2 ( t + ν 2 - 1 ) ) , for t ∈ [ 0 , b ] N 0 , subject to y i ( ν i  − 2) =  ψ i ( y i ) and y i ( ν i  +  b ) =  ϕ i ( y i ), for i  = 1, 2, where ψ i , ϕ 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 ( y 1 ,  y 2 ) are nonnegative for each i , we do not necessarily presume that each ψ i ( y i ) and ϕ 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.