Banach operator pairs and common fixed points in hyperconvex metric spaces (Q640118)

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