A Generalization of Suzuki's Lemma. (15th June 2011) - Panyanak, B.; Cuntavepanit, A.
Abstract : Let{ z n }, { w n }, and{ v n } be bounded sequences in a metric space of hyperbolic type( X, d ), and let{ α n } be a sequence in[ 0, 1 ] with0 < lim     inf   n   α n ≤ lim     sup   n   α n < 1 . Ifz n + 1 = α n w n ⊕ ( 1 − α n ) v n for alln ∈ ℕ, lim   n   d ( z n, v n ) = 0, andlim     sup   n   ( d ( w n + 1, w n ) - d ( z n + 1, z n ) ) ≤ 0, thenlim   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.