Abstract and Applied Analysis

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

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Abstract

A discrete equation Δ y ( 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 : n 0 - k ( 0 , ) , where n 0 = { n 0 , n 0 + 1 , } , n 0 and n n 0 . 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.

Article information

Source
Abstr. Appl. Anal., Volume 2011, Number 1 (2011), Article ID 709427, 15 pages.

Dates
First available in Project Euclid: 12 August 2011

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1313171420

Digital Object Identifier
doi:10.1155/2011/709427

Mathematical Reviews number (MathSciNet)
MR2817261

Zentralblatt MATH identifier
1220.39004

Citation

Berezansky, L.; Diblík, J.; Růžičková, M.; Šutá, Z. Asymptotic Convergence of the Solutions of a Discrete Equation with Two Delays in the Critical Case. Abstr. Appl. Anal. 2011 (2011), no. 1, Article ID 709427, 15 pages. doi:10.1155/2011/709427. https://projecteuclid.org/euclid.aaa/1313171420


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