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Mathematical Problems in Engineering
Volume 2015, Article ID 836283, 14 pages
Research Article

On Some Boundedness and Convergence Properties of a Class of Switching Maps in Probabilistic Metric Spaces with Applications to Switched Dynamic Systems

1Institute of Research and Development of Processes, University of the Basque Country, Campus of Leioa (Biscay), P.O. Box 644, Bilbao, Barrio Sarriena, 48940 Leioa, Spain
2Department of Telecommunications and Systems Engineering, Universitat Autònoma de Barcelona (UAB), Bellaterra, Cerdanyola del Vallès, 08193 Barcelona, Spain

Received 10 June 2015; Accepted 9 September 2015

Academic Editor: Wenguang Yu

Copyright © 2015 M. De la Sen and A. Ibeas. 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.


This paper investigates some boundedness and convergence properties of sequences which are generated iteratively through switched mappings defined on probabilistic metric spaces as well as conditions of existence and uniqueness of fixed points. Such switching mappings are built from a set of primary self-mappings selected through switching laws. The switching laws govern the switching process in between primary self-mappings when constructing the switching map. The primary self-mappings are not necessarily contractive but if at least one of them is contractive then there always exist switching maps which exhibit convergence properties and have a unique fixed point. If at least one of the self-mappings is nonexpansive or an appropriate combination given by the switching law is nonexpansive, then sequences are bounded although not convergent, in general. Some illustrative examples are also given.