Relaxed viscosity approximation methods for fixed point problems and variational inequality problems

Let X be a real strictly convex and reflexive Banach space with a uniformly Gateaux differentiable norm and C be a nonempty closed convex subset of X. Let {T-n}(n=1)(infinity) be a sequence of nonexpansive self-mappings on C such that the common fixed point set F := boolean AND(infinity)(n=1) F(T-n) not equal circle divide and f : C -> C be a given contractive mapping, and {lambda(n)} be a sequence of nonnegative numbers in [0, 1]. Consider the following relaxed viscosity approximation method {x(n+1) = (1 - alpha(n) - beta(n))x(n) + alpha(n) f(y(n)) + beta(n) W(n)y(n). y(n) = (1 - gamma(n))x(n) + gamma(n) W(n)x(n), n >= 1 where W-n is the W-mapping generated by T-n, Tn-1,..., T-1 and lambda(n,) lambda(n-1),..., lambda(1) for each n >= 1. It is proven that under very mild conditions on the parameters, the sequence {x(n)} of approximate solutions generated by the proposed method converges strongly to some p is an element of F where p is the unique solution in F to the following variational inequality: <( 1 - f) p, J(p - x(*))> <= 0. x(*) is an element of F. (c) 2007 Elsevier Ltd. All rights reserved.