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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 [{\it R. DeMarr}, Pac. J. Math. 13, 1139--1141 (1963; Zbl 0191.14901)]. It is based on the very recent work of {\it J.-R. Chen} and {\it 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.

54H25Fixed-point and coincidence theorems in topological spaces
47H20Semigroups of nonlinear operators
47H09Mappings defined by “shrinking” properties
54E40Special maps on metric spaces
41A65Abstract approximation theory
Full Text: DOI
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