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.

Abstract

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:K→K be a uniformly continuous pseudocontraction. If f:K→K 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 z1∈K, zn+1=μn(αnTzn+(1−αn)zn)+(1−μn)f(zn), n∈ℕ, strongly converges to the fixed point of T, which is the unique solution of some variational inequality, provided that K is bounded.