Abstract
Let K be a nonempty closed and convex subset of a real Banach space E. Let T : K -> E be a nonexpansive weakly inward mapping with F(T) not equal empty set; and f : K -> K be a contraction. Then for t is an element of (0, 1), there exists a sequence {y(t)} subset of K satisfying y(t) = (1 - t)f(y(t)) + tT(y(t)). Furthermore, if E is a strictly convex real reflexive Banach space having a uniformly Gateaux differentiable norm, then {y(t)} converges strongly to a fixed point p of T such that p is the unique solution in F(T) to a certain variational inequality. Moreover, if {T-i; i = 1, 2, ... r} is a family of nonexpansive mappings, then an explicit iteration process which converges strongly to a common fixed point of {T-i, i = 1, 2, ..., r} and to a solution of a certain variational inequality is constructed. Under the above setting, the family T-i, i = 1, 2, ..., r need not satisfy the requirement that (r)boolean AND(i=1) F(T-i) = F(T-r Tr-1 ,..., T-1) = F(T-1 T-r ,..., T-2) = ,..., = F(Tr-1 Tr-2 ,..., T1Tr). (C) 2007 Published by Elsevier Inc.