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On approximating curves associated with nonexpansive mappings. (English) Zbl 1265.54208
Summary: Let $$X$$ be a Banach space with metric $$d$$. Let $$T,N:X\to X$$ be a strict $$d$$-contraction and a $$d$$-nonexpansive map, respectively. In this paper we investigate the properties of the approximating curve associated with $$T$$ and $$N$$. Moreover, following M. Levenshtein and S. Reich [Nonlinear Anal., Theory Methods Appl. 70, No. 12, A, 4145–4150 (2009; Zbl 1176.58004)], we consider the approximating curve associated with a holomorphic map $$f:B\to\alpha B$$ and a $$\rho$$-nonexpansive map $$M:B\to B$$, where $$B$$ is the open unit ball of a complex Hilbert space $$H$$, $$\rho$$ is the hyperbolic metric defined on $$B$$ and $$0\leq\alpha < 1$$. We give conditions on $$f$$ and $$M$$ for this curve to be injective, and we show that this curve is continuous.

MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects) 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47H10 Fixed-point theorems