A generalization of Suzuki's lemma

Abstract

Let {zn}, {wn}, and {vn} be bounded sequences in a metric space of hyperbolic type (X, d), and let {αn} be a sequence in [0,1] with 0 < lim infnαn< lim supnαn< 1. If zn+1=αnwn(1-αn)vnfor all n ∈ ℕ , limnd (zn, vn) = 0, and lim supn(d (wn+1, wn) - d (zn+1, zn)) ≤ 0, then limnd (wn, zn) = 0. This is a generalization of Lemma 2.2 in (T. Suzuki, 2005). As a consequence, we obtain strong convergence theorems for the modified Halpern iterations of nonexpansive mappings in CAT(0) spaces. Copyright © 2011 B. Panyanak and A. Cuntavepanit.