On positive solutions to nonlocal fractional and integer-order difference equations

Abstract

In this paper, we consider a discrete fractional boundary value problem of the form -Δνy(t) = f(t + ν - 1, y(t + ν - 1)), y(ν - 2) = ψ(y), y(ν + b) = φ(y), where t ∈ [0,b]N0, f: [ν/-1, ...,ν +6-1]Nν-2 × R{double-struck} → [0,+∞) is continuous, ψ, φ: C([ν - 2, ν + b]Nν-2) → R{double-struck} are given functionals, and 1 < ν ≤ 2. We show that provided that both ψ and φ are linear functionals, then under certain conditions the fractional boundary value problem will have at least one positive solution even if neither ψ nor φ is nonnegative for all y ≥ 0. This provides new results not only for the fractional boundary value problem but also in the case when ν = 2. Our results also generalize some recent work on the conjugate fractional boundary value problem. We conclude with two examples to illustrate our results.