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Journal of Function Spaces
Volume 2015, Article ID 470574, 11 pages
Research Article

Some Results on Best Proximity Points of Cyclic Contractions in Probabilistic Metric Spaces

1Institute of Research and Development of Processes IIDP, Faculty of Science and Technology, University of the Basque Country, P.O. Box 644 de Bilbao Barrio Sarriena, Leioa, 48940 Bizkaia, Spain
2Department of Mathematics, ATILIM University, Incek, 06836 Ankara, Turkey
3Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University, Jeddah 21589, Saudi Arabia

Received 12 October 2014; Accepted 30 December 2014

Academic Editor: Calogero Vetro

Copyright © 2015 Manuel De la Sen and Erdal Karapınar. 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 properties of convergence of distances of -cyclic contractions on the union of the subsets of an abstract set defining probabilistic metric spaces and Menger probabilistic metric spaces as well as the characterization of Cauchy sequences which converge to the best proximity points. The existence and uniqueness of fixed points and best proximity points of -cyclic contractions defined in induced complete Menger spaces are also discussed in the case when the associate complete metric space is a uniformly convex Banach space. On the other hand, the existence and the uniqueness of fixed points of the -composite mappings restricted to each of the subsets in the cyclic disposal are also investigated and some illustrative examples are given.