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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 370982, 14 pages
http://dx.doi.org/10.1155/2011/370982
Research Article

Weighted Asymptotically Periodic Solutions of Linear Volterra Difference Equations

1Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, 66237 Brno, Czech Republic
2Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, 61600 Brno, Czech Republic
3Department of Mathematics, University of Žilina, 01026 Žilina, Slovakia
4Faculty of Electrical Engineering, Institute of Mathematics, Poznań University of Technology, 60965 Poznań, Poland

Received 16 January 2011; Accepted 17 March 2011

Academic Editor: Elena Braverman

Copyright © 2011 Josef 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

A linear Volterra difference equation of the form ๐‘ฅ(๐‘›+1)=๐‘Ž(๐‘›)+๐‘(๐‘›)๐‘ฅ(๐‘›)+โˆ‘๐‘›๐‘–=0๐พ(๐‘›,๐‘–)๐‘ฅ(๐‘–), where ๐‘ฅโˆถโ„•0โ†’โ„, ๐‘Žโˆถโ„•0โ†’โ„, ๐พโˆถโ„•0ร—โ„•0โ†’โ„ and ๐‘โˆถโ„•0โ†’โ„โงต{0} is ๐œ”-periodic, is considered. Sufficient conditions for the existence of weighted asymptotically periodic solutions of this equation are obtained. Unlike previous investigations, no restriction on โˆ๐œ”โˆ’1๐‘—=0๐‘(๐‘—) is assumed. The results generalize some of the recent results.