Fixed Point Theory and Applications
Volume 2011 (2011), Article ID 712706, 22 pages
Research Article

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

Department of Nonlinear Analysis, Faculty of Mathematics and Computer Science, University of Łódź, Banacha 22, 90-238 Łódź, Poland

Received 13 September 2010; Revised 19 October 2010; Accepted 10 November 2010

Academic Editor: Jen Chih Yao

Copyright © 2011 Kazimierz Włodarczyk and Robert Plebaniak. 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.


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 𝐾 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.