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Fixed points of uniformly Lipschitzian mappings. (English) Zbl 1105.47050
The authors study and compare two fixed-point theorems for uniformly Lipschitzian maps, due respectively to E. A. Lifšic [Voronezh. Gos. Univ. Trudy Mat. Fak. Vyp. 16 Sb. Statei po Nelineinym Operator. Uravn. i Prilozhen., 23–28 (1975; MR 57 #17401)]and to T.–C. Lim and H.–K. Xu [Nonlinear Anal., Theory Methods Appl. 25, No. 11, 1231–1235 (1995; Zbl 0845.47045)], taking for an underlying framework the so-called CAT(0)-spaces. They show that both these results fit into this setting. This an important contribution of the paper since it provides probably the first example of a class of spaces which are not Banach spaces, but for which the Lifšic characteristic may be computed. It appears that, in this setting, the Lifšic result is sharper. Next, the authors introduce a new property, weaker than property (P) of Lim and Xu, that yields fixed-points for uniformly Lipschitzian mappings in a class of hyperconvex spaces, a class which includes those possessing property (P).

MSC:
47H10Fixed point theorems for nonlinear operators on topological linear spaces
54H25Fixed-point and coincidence theorems in topological spaces
54E40Special maps on metric spaces
47H09Mappings defined by “shrinking” properties