Fixed Point Theory and Applications
Volume 2007 (2007), Article ID 78628, 8 pages
doi:10.1155/2007/78628
Research Article
An Extension of Gregus Fixed Point Theorem
Mathematics Department, University of Lagos, P.O. Box 31, Lagos, Nigeria
Received 2 October 2006; Accepted 17 December 2006
Academic Editor: Lech Gorniewicz
Copyright © 2007 J. O. Olaleru and H. Akewe. 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
Let C be a closed convex subset of a complete metrizable topological vector space (X,d) and T:C→C a mapping that satisfies d(Tx,Ty)≤ad(x,y)+bd(x,Tx)+cd(y,Ty)+ed(y,Tx)+fd(x,Ty) for all x,y∈C, where 0<a<1, b≥0, c≥0, e≥0, f≥0, and a+b+c+e+f=1. Then T has a unique fixed point. The above theorem, which is a generalization and an extension of the results of several authors, is proved in this paper. In addition, we use the Mann iteration to approximate the fixed point of T.