Large-time behavior of a class of positive solutions of discrete equation Δu(n + k) = −p(n)u(n) in the critical case

Abstract

It is well-known that the discrete delayed equation Δu(n+k)=−pcu(n), where k is a positive integer and pc=kk(k+1)k+1 has a positive solution u = u(n), n = 0, 1, 2,…. This is no longer true for the equation Δu(n+k)=−pu(n) where the constant p > pc. In the paper, we study the delayed discrete equation Δu(n+k)=−p∗(n)u(n) with a function p∗(n) positive for all sufficiently large n. This function has a special form and satisfies the inequality p∗(n) > pc. We prove that, even in this case, there exists a class of positive solutions of n → ∞ and derive two-sided estimates characterizing their behavior.