Journal of Applied Mathematics
Volume 2011 (2011), Article ID 542941, 17 pages
http://dx.doi.org/10.1155/2011/542941
Research Article

On a General Contractive Condition for Cyclic Self-Mappings

M. De la Sen

Institute of Research and Development of Processes, Faculty of Science and Technology, University of the Basque Country, Campus de Leioa (Bizkaia), Apartado 644 de Bilbao, 48080 Bilbao, Spain

Received 26 April 2011; Accepted 3 October 2011

Academic Editor: J. C. Butcher

Copyright © 2011 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.

Abstract

This paper is concerned with 𝑝 ( ≥ 2 ) -cyclic self-mappings 𝑇 ∶ ⋃ 𝑖 ∈ 𝑝 𝐴 𝑖 → ⋃ 𝑖 ∈ 𝑝 𝐴 𝑖 in a metric space ( 𝑋 , 𝑑 ), with 𝐴 𝑖 ⊂ 𝑋 , 𝑇 ( 𝐴 𝑖 ) ⊆ 𝐴 𝑖 + 1 for 𝑖 = 1 , 2 , … , 𝑝 , under a general contractive condition which includes as particular cases several of the existing ones in the literature. The existence and uniqueness of fixed points and best proximity points is discussed as well as the convergence to them of the iterates generated by the self-mapping from given initial points.