Approximation Methods for a Common Fixed Point of a Finite Family of Nonexpansive Mappings

Abstract

Let K be a nonempty closed and convex subset of a real Banach space E. Let T: K → E be a continuous pseudocontractive mapping and f:K → E a contraction, both satisfying weakly inward condition. Then for t ∈ (0, 1), there exists a sequence {y t } ⊂ K satisfying the following condition: y t = (1 − t)f(y t ) + tT(y t ). Suppose further that {y t } is bounded or F(T) ≠ ∅ and E is a reflexive Banach space having weakly continuous duality mapping J ϕ for some gauge ϕ. Then it is proved that {y t } converges strongly to a fixed point of T, which is also a solution of certain variational inequality. Moreover, an explicit iteration process that converges strongly to a common fixed point of a finite family of nonexpansive mappings and hence to a solution of a certain variational inequality is constructed.