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

Abstract

A discrete equationΔy(n)=β(n)[y(n−j)−y(n−k)]with two integer delayskandj, k>j≥0is considered forn→∞. We assumeβ:ℤn0−k∞→(0,∞), whereℤn0∞={n0,n0+1,…},  n0∈ℕandn∈ℤn0∞. Criteria for the existence of strictly monotone and asymptotically convergent solutions forn→∞are presented in terms of inequalities for the functionβ. Results are sharp in the sense that the criteria are valid even for some functionsβwith 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.