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Banach operator pairs and common fixed points in hyperconvex metric spaces. (English) Zbl 1235.54037

This paper is concerned with de Marr’s result on the existence of a common fixed point for an arbitrary family of symmetric Banach operator pairs in hyperconvex metric spaces without assuming compactness [R. DeMarr, Pac. J. Math. 13, 1139–1141 (1963; Zbl 0191.14901)]. It is based on the very recent work of J.-R. Chen and Z.-K. Li [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 10, 3086–3090 (2011; Zbl 1252.47054)]. Necessary and sufficient conditions for an invertible semigroup of isometric mappings in hyperconvex metric spaces to have a common fixed point are given. Moreover, some results on invariant approximations for Banach operator pairs in hyperconvex metric spaces are also discussed.

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
47H20 Semigroups of nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
54E40 Special maps on metric spaces
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
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References:

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