ASYMPTOTIC NONUNIFORM NONRESONANCE CONDITIONS FOR A NONLINEAR DISCRETE BOUNDARY VALUE PROBLEM

Let T := {a+1, . . . , b+1}. We study the solvability of nonlinear discrete two-point boundary value problem ( ∆2u(t − 1) + g(t, u(t)) = h(t), t ∈ T, u(a) = u(b + 2) = 0 where h : T → R, g : T × R → R satisfies α(t) ≤ lim inf |x|→∞ x −1 g(t, x) ≤ lim sup |x|→∞ x −1 g(t, x) ≤ β(t) uniformly on T, 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 LeraySchauder continuation theorem.