Asymptotic nonuniform nonresonance conditions for a nonlinear discrete boundary value problem

Let double struck T sign := {α + 1, ..., b + 1}. We study the solvability of nonlinear discrete two-point boundary value problem {δ²u(t - 1) + g(t, u(t)) = h(t), t ∈ double struck T sign, u(a) = u(b + 2) = 0 where h :double struck T sign → ℝ, g : double struck T sign × ℝ → ℝ satisfies α(t) ≤ lim inf_|x|→∞ x^-1 g(t, x) ≤ lim sup _|x|→∞ x^-1g(t, x) ≤ β(t) uniformly on double struck T sign, and α and β satisfy some nonresonance conditions of nonuniform type with respect to two consecutive eigenvalues of the associated linear problem. The proof is based on the Leray-Schauder continuation theorem.