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⁡⁡inf⁡n⁡αn≤lim⁡⁡sup⁡n⁡αn<1. If zn+1=αnwn⊕(1−αn)vn for all n∈ℕ, lim⁡n⁡d(zn,vn)=0, and lim⁡⁡sup⁡n⁡(d(wn+1,wn)-d(zn+1,zn))≤0, then lim⁡n⁡d(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.