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On the Schauder fixed point theorem. (English) Zbl 0852.55005
Gȩba, Kazimierz (ed.) et al., Topology in nonlinear analysis. Papers of two minisemesters, Warsaw, Poland, Fall 1994. Warszawa: Polish Academy of Sciences, Inst. of Mathematics, Banach Cent. Publ. 35, 207-219 (1996).
The famous Schauder fixed point theorem proved in 1930 says that any continuous and compact map of a convex and closed subset of a Banach space has a fixed point. This paper contains a survey of various results concerning this theorem for metric spaces both in single-valued and multi-valued cases. A number of open problems is formulated.
For the entire collection see [Zbl 0840.00029].
Reviewer: Z.Čerin (Zagreb)

55M20 Fixed points and coincidences in algebraic topology
54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems
47H04 Set-valued operators