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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 310106, 12 pages
Research Article

Some Results on Fixed and Best Proximity Points of Precyclic Self-Mappings

Institute of Research and Development of Processes, University of Basque Country, Campus of Leioa (Bizkaia)-Aptdo. Postal 644-Bilbao, 48080 Bilbao, Spain

Received 11 March 2013; Accepted 2 July 2013

Academic Editor: D. R. Sahu

Copyright © 2013 M. De la Sen. 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 is devoted to investigating the limit properties of distances and the existence and uniqueness of fixed points, best proximity points and existence, and uniqueness of limit cycles, to which the iterated sequences converge, of single-valued, and so-called, contractive precyclic self-mappings which are proposed in this paper. Such self-mappings are defined on the union of a finite set of subsets of uniformly convex Banach spaces under generalized contractive conditions. Each point of a subset is mapped either in some point of the same subset or in a point of the adjacent subset. In the general case, the contractive condition of contractive precyclic self-mappings is admitted to be point dependent and it is only formulated on a complete disposal, rather than on each individual subset, while it involves a condition on the number of iterations allowed within each individual subset before switching to its adjacent one. It is also allowed that the distances in-between adjacent subsets can be mutually distinct including the case of potential pairwise intersection for only some of the pairs of adjacent subsets.