A Generalization of Suzuki's Lemma

Abstract EN

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

If zn+1=αnwn⊕(1−αn)vn for all n∈ℕ, limnd(zn,vn)=0, and limsupn(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.