Abstract and Applied Analysis
Volume 2012 (2012), Article ID 580750, 20 pages
http://dx.doi.org/10.1155/2012/580750
Research Article

Asymptotic Convergence of the Solutions of a Dynamic Equation on Discrete Time Scales

J. Diblík,1,2 M. Růžičková,3 Z. Šmarda,1 and Z. Šutá3

1Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, 616 00 Brno, Czech Republic
2Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, 602 00 Brno, Czech Republic
3Department of Mathematics, University of Žilina, 01026 Žilina, Slovakia

Received 10 September 2011; Accepted 12 November 2011

Academic Editor: Toka Diagana

Copyright © 2012 J. Diblík 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

The paper investigates a dynamic equation Δ 𝑦 ( 𝑡 𝑛 ) = 𝛽 ( 𝑡 𝑛 ) [ 𝑦 ( 𝑡 𝑛 − 𝑗 ) − 𝑦 ( 𝑡 𝑛 − 𝑘 ) ] for 𝑛 → ∞ , where 𝑘 and 𝑗 are integers such that 𝑘 > 𝑗 ≥ 0 , on an arbitrary discrete time scale 𝕋 ∶ = { 𝑡 𝑛 } with 𝑡 𝑛 ∈ ℝ , 𝑛 ∈ ℤ ∞ 𝑛 0 − 𝑘 = { 𝑛 0 − 𝑘 , 𝑛 0 − 𝑘 + 1 , … } , 𝑛 0 ∈ ℕ , 𝑡 𝑛 < 𝑡 𝑛 + 1 , Δ 𝑦 ( 𝑡 𝑛 ) = 𝑦 ( 𝑡 𝑛 + 1 ) − 𝑦 ( 𝑡 𝑛 ) , and l i m 𝑛 → ∞ 𝑡 𝑛 = ∞ . We assume 𝛽 ∶ 𝕋 → ( 0 , ∞ ) . It is proved that, for the asymptotic convergence of all solutions, the existence of an increasing and asymptotically convergent solution is sufficient. Therefore, the main attention is paid to the criteria for the existence of an increasing solution asymptotically convergent for 𝑛 → ∞ . The results are presented as inequalities for the function 𝛽 . Examples demonstrate that the criteria obtained are sharp in a sense.