A Generalization of Suzuki's Lemma

Abstract

Let $\left\{z_n\right\}$, $\left\{w_n\right\}$, and $\left\{v_n\right\}$ be bounded sequences in a metric space of hyperbolic type $(X,d)$, and let $\left\{{\alpha }_n\right\}$ be a sequence in $[0,1]$ with $0<{\lim \inf }_n{\alpha }_n\le {\lim \sup }_n{\alpha }_n<1$. If $z_{n+1}={\alpha }_n{w}_n\oplus (1-{\alpha }_n)v_n$ for all $n\in \mathbb{N}$, ${\lim }_{n}d(z_n,v_n)=0$ , and ${\lim \sup }_n(d(w_{n+1},w_n)-d(z_{n+1},z_n\left)\right)\le 0$ , then ${\lim }_{n}d(w_n,z_n)=0$. 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.