Let D be a closed convex subset
of a Banach space X, let T : D → D be nonexpansive (that is, ∥Tx−Ty∥≦∥x−y∥
for every x,y ∈ D), and let Fλ= λT + (1 − λ)I, where λ ∈ (0,1) and I denotes the
identity on D. Several authors have found conditions under which the sequences of
iterates {Tnx}, or the sequences {Fλnx}, converge strongly or weakly to
fixed points of T for all x ∈ D. In this paper we establish conditions under
which the sequences {F1∕2nx} converge strongly to fixed points of T for all x
in a neighborhood of the fixed point set of T; furthermore, our theorems
hold for classes of mappings T more general than the class of nonexpansive
mappings.
