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Strong convergence of modified Halpern iterations in \(\mathrm{CAT}(0)\) spaces. (English) Zbl 1206.47072

Summary: Strong convergence theorems are established for the modified Halpern iterations of nonexpansive mappings in CAT(0) spaces. Our results extend and improve the recent ones announced by T.-H.Kim and H.-K.Xu [Nonlinear Anal., Theory Methods Appl.61, No. 1–2, A, 51–60 (2005; Zbl 1091.47055)], L.-G.Hu [Fixed Point Theory Appl.2008, Article ID 649162 (2008; Zbl 1203.47049)], Y.-S.Song and R.-D.Chen [Math.Nachr.281, No.8, 1196–1204 (2008; Zbl 1220.47121)], S.Saejung [Fixed Point Theory Appl.2010, Article ID 471781 (2010; Zbl 1197.54074)], and many others.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
54H25 Fixed-point and coincidence theorems (topological aspects)
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