Open Access

Quasigauge Spaces with Generalized Quasipseudodistances and Periodic Points of Dissipative Set-Valued Dynamic Systems

Fixed Point Theory and Applications20102011:712706

https://doi.org/10.1155/2011/712706

Received: 13 September 2010

Accepted: 10 November 2010

Published: 25 November 2010

Abstract

In quasigauge spaces, we introduce the families of generalized quasipseudodistances, and we define three kinds of dissipative set-valued dynamic systems with these families of generalized quasi-pseudodistances and with some families of not necessarily lower semicontinuous entropies and next, assuming that quasigauge spaces are left sequentially complete (but not necessarily Hausdorff), we prove that for each starting point each dynamic process or generalized sequence of iterations of these dissipative set-valued dynamic systems left converges and we also show that if an iterate of these dissipative set-valued dynamic systems is left quasiclosed, then these limit points are periodic points. Examples illustrating ideas, methods, definitions, and results are constructed.

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Authors’ Affiliations

(1)
Department of Nonlinear Analysis, Faculty of Mathematics and Computer Science, University of Łódź

Copyright

© Kazimierz Włodarczyk and Robert Plebaniak. 2011

This article is published under license to BioMed Central Ltd. 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.