Fixed sign solutions of second-order difference equations with Neumann boundary conditions

In this paper, we study the existence of solution of the nonlinear second-order difference problem with Neumann boundary conditions. Assuming the existence of a pair of ordered lower and upper solutions γ and β, we obtain optimal existence results for the case γ ≤ β and even for γ ≥ β. To this end, we do an exhaustive study of the values of the real parameters α and μ, for which the associated Greenʹs function to the linear operator L[α,μ] uκ ≡ uκ+2 − 2 α uκ+1 + μ uκ with Neumann boundary conditions has fixed sign.