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A generalization of Suzuki’s Lemma. (English) Zbl 1221.54065

Summary: Let \(\{z_n\}\), \(\{w_n\}\), and \(\{v_n\}\), be bounded sequences in a metric space of hyperbolic type \((X,d)\), and let \(\{\alpha_n\}\) be a sequence in \([0,1]\) with \(0<\liminf_n\alpha_n\leq\limsup_n\alpha_n<1\). If \(z_{n+1}=\alpha_nw_n\oplus (1-\alpha_n)v_n\) for all \(n\in\mathbb N\), \(\lim_nd(z_n,v_n)=0\), and \(\limsup_n (d(w_{n+1},w_n) -d(z_{n+1},z_n))\leq 0\), then \(\lim_n d(w_n,z_n)=0\). This is a generalization of Lemma 2.2 in [T. Suzuki, Fixed Point Theory Appl. 2005, No. 1, 103–123 (2005; Zbl 1123.47308)]. As a consequence, we obtain strong convergence theorems for the modified Halpern iterations of nonexpansive mappings in CAT(0) spaces.

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47J25 Iterative procedures involving nonlinear operators

Citations:

Zbl 1123.47308
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References:

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