Asymptotic convergence of the solutions of a discrete equation with two delays in the critical case

A discrete equation Δy(n) = β(n)[y(n-j)-y(n-k)] with two integer delays k and j, k > j ≥ 0 is considered for n → ∞. We assume β : ℤ_n0-k^∞ → (0, ∞), where ℤ_n0^∞ = {n0, n0 + 1,}, n0 ∈ ℕ and n ∈ ℤ_n0^∞. Criteria for the existence of strictly monotone and asymptotically convergent solutions for n → ∞ are presented in terms of inequalities for the function β. Results are sharp in the sense that the criteria are valid even for some functions β with a behavior near the so-called critical value, defined by the constant (k-j)^-1. Among others, it is proved that, for the asymptotic convergence of all solutions, the existence of a strictly monotone and asymptotically convergent solution is sufficient.