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Abstract and Applied Analysis
Volume 2014, Article ID 627295, 37 pages
Research Article

General Explicit Solution of Planar Weakly Delayed Linear Discrete Systems and Pasting Its Solutions

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

Received 3 September 2013; Accepted 21 October 2013; Published 29 April 2014

Academic Editor: Miroslava Růžičková

Copyright © 2014 Josef Diblík and Hana Halfarová. 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.


Planar linear discrete systems with constant coefficients and delays are considered where , are constant integer delays, , are constant matrices, and . It is assumed that the considered system is weakly delayed. The characteristic equations of such systems are identical with those for the same systems but without delayed terms. In this case, after several steps, the space of solutions with a given starting dimension is pasted into a space with a dimension less than the starting one. In a sense, this situation is analogous to one known in the theory of linear differential systems with constant coefficients and special delays when the initially infinite dimensional space of solutions on the initial interval turns (after several steps) into a finite dimensional set of solutions. For every possible case, explicit general solutions are constructed and, finally, results on the dimensionality of the space of solutions are obtained.