A Generalization of Suzuki′s Lemma
Bancha Panyanak, Asawathep Cuntavepanit · Abstract and Applied Analysis · 2011
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. Ifzn+1=αnwn⊕ (1 −αn)vnfor alln∈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.