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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.

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