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.
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