Abstract and Applied Analysis
Volume 2011 (2011), Article ID 928587, 12 pages
http://dx.doi.org/10.1155/2011/928587
Research Article

Periodic Problems of Difference Equations and Ergodic Theory

B. A. Biletskyi,1 A. A. Boichuk,2 and A. A. Pokutnyi2

1Department of Inductive Modeling and Control Glushkov, Institute of Cybernetics NAS Ukraine, No. 40 Glushkov Avenue, Kyiv 03680, Ukraine
2Laboratory of boundary value problems of differential equations theory, Institute of Mathematics of NAS of Ukraine, No. 3 Tereshenkivska Street, Kyiv 01601, Ukraine

Received 1 February 2011; Accepted 11 May 2011

Academic Editor: Miroslava Růžičková

Copyright © 2011 B. A. Biletskyi 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 necessary and sufficient conditions for solvability of the family of difference equations with periodic boundary condition were obtained using the notion of relative spectrum of linear bounded operator in the Banach space and the ergodic theorem. It is shown that when the condition of existence is satisfied, then such periodic solutions are built using the formula for the generalized inverse operator to the linear limited one.