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

L. Berezansky, J. Diblík, M. Růžičková and Z. Šutá

Abstract and Applied Analysis, 2011, vol. 2011, issue 1

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.

Date: 2011
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https://doi.org/10.1155/2011/709427

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnlaaa:v:2011:y:2011:i:1:n:709427

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