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

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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.