Asymptotic Convergence of the Solutions of a Discrete Equation with Two Delays in the Critical Case

Abstract

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:ℤn0−k∞→(0,∞), where ℤn0∞={n0,n0+1,…}, n0∈ℕ and n∈ℤn0∞. 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.