Approximation of fixed points of weakly contractive nonself maps in Banach spaces

Let K be a closed convex subset of a real uniformly smooth Banach space E. Suppose K is a nonexpansive retract of E with P as the nonexpansive retraction. Let T :K →E be a d-weakly contractive map such that a fixed point x∗ ∈ int(K) of T exists. It is proved that a descent-like approximation sequence converges strongly to x∗. Furthermore, if K is a nonempty closed convex subset of an arbitrary real Banach space and T :K →K is a uniformly continuous d-weakly contractive map with F(T ) := {x ∈ K: T x = x} = ∅, it is proved that a descent-like approximation sequence converges strongly to x∗ ∈ F(T ).  2002 Elsevier Science (USA). All rights reserved