Fixed Point Theory and Applications
Volume 2006 (2006), Article ID 69758, 12 pages

Fixed point variational solutions for uniformly continuous pseudocontractions in Banach spaces

Aniefiok Udomene

Department of Mathematics, Statistics, & Computer Science, University of Port Harcourt, Port Harcourt PMB 5323, Nigeria

Received 27 June 2005; Revised 21 November 2005; Accepted 28 November 2005

Copyright © 2006 Aniefiok Udomene. 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.


Let E be a reflexive Banach space with a uniformly Gâteaux differentiable norm, let K be a nonempty closed convex subset of E, and let T:KK be a uniformly continuous pseudocontraction. If f:KK is any contraction map on K and if every nonempty closed convex and bounded subset of K has the fixed point property for nonexpansive self-mappings, then it is shown, under appropriate conditions on the sequences of real numbers {αn}, {μn}, that the iteration process z1K, zn+1=μn(αnTzn+(1αn)zn)+(1μn)f(zn), n, strongly converges to the fixed point of T, which is the unique solution of some variational inequality, provided that K is bounded.