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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 709427, 15 pages
doi:10.1155/2011/709427
Research Article

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

L. Berezansky,1 J. Diblík,2 M. Růžičková,2 and Z. Šutá2

1Department of Mathematics, Ben-Gurion University of the Negev, 84105 Beer-Sheva, Israel
2Department of Mathematics, University of Žilina, 01026 Žilina, Slovakia

Received 9 October 2010; Revised 17 March 2011; Accepted 13 April 2011

Academic Editor: Elena Braverman

Copyright © 2011 L. Berezansky et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A discrete equation Ξ” 𝑦 ( 𝑛 ) = 𝛽 ( 𝑛 ) [ 𝑦 ( 𝑛 βˆ’ 𝑗 ) βˆ’ 𝑦 ( 𝑛 βˆ’ π‘˜ ) ] with two integer delays π‘˜ and 𝑗 , π‘˜ > 𝑗 β‰₯ 0 is considered for 𝑛 β†’ ∞ . We assume 𝛽 ∢ β„€ ∞ 𝑛 0 βˆ’ π‘˜ β†’ ( 0 , ∞ ) , where β„€ ∞ 𝑛 0 = { 𝑛 0 , 𝑛 0 + 1 , … } , 𝑛 0 ∈ β„• and 𝑛 ∈ β„€ ∞ 𝑛 0 . Criteria for the existence of strictly monotone and asymptotically convergent solutions for 𝑛 β†’ ∞ 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 ( π‘˜ βˆ’ 𝑗 ) βˆ’ 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.