TY - JOUR

T1 - Asymptotic convergence of the solutions of a discrete equation with two delays in the critical case

AU - Diblík, J.

AU - Berezansky, L.

AU - Růžičková, M.

AU - Šutá, Z.

PY - 2011/9/16

Y1 - 2011/9/16

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=80052688643&partnerID=8YFLogxK

U2 - 10.1155/2011/709427

DO - 10.1155/2011/709427

M3 - Article

AN - SCOPUS:80052688643

VL - 2011

JO - Abstract and Applied Analysis

JF - Abstract and Applied Analysis

SN - 1085-3375

M1 - 709427

ER -