Abstract and Applied Analysis
Volume 2011 (2011), Article ID 586328, 28 pages
http://dx.doi.org/10.1155/2011/586328
Research Article

A Final Result on the Oscillation of Solutions of the Linear Discrete Delayed Equation Δ 𝑥 ( 𝑛 ) = − 𝑝 ( 𝑛 ) 𝑥 ( 𝑛 − 𝑘 ) with a Positive Coefficient

J. Baštinec,1 L. Berezansky,2 J. Diblík,1,3 and Z. Šmarda1

1Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, 61600 Brno, Czech Republic
2Department of Mathematics, Ben-Gurion University of the Negev, 84105 Beer-Sheva, Israel
3Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, 602 00 Brno, Czech Republic

Received 15 January 2011; Accepted 6 June 2011

Academic Editor: Yuri V. Rogovchenko

Copyright © 2011 J. Baštinec 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 linear ( 𝑘 + 1 ) th-order discrete delayed equation Δ 𝑥 ( 𝑛 ) = − 𝑝 ( 𝑛 ) 𝑥 ( 𝑛 − 𝑘 ) where 𝑝 ( 𝑛 ) a positive sequence is considered for 𝑛 → ∞ . This equation is known to have a positive solution if the sequence 𝑝 ( 𝑛 ) satisfies an inequality. Our aim is to show that, in the case of the opposite inequality for 𝑝 ( 𝑛 ) , all solutions of the equation considered are oscillating for 𝑛 → ∞ .