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Almost fixed points of multimaps having totally bounded ranges. (English) Zbl 1005.47054
The main thrust of this paper centres around a fixed point theorem of A. Idzik [Proc. Am. Math. Soc. 104, No. 3, 779-784 (1988; Zbl 0691.47046)] for a Kakutani map. In the author’s words: “We give almost fixed point theorems for Kakutani maps or for a larger class of multimaps (so called, the better admissible class) having totally bounded ranges. Precisely, we assume that the closures of the ranges satisfy more restrictive conditions than that of convexly totally bounded sets. Our results are applied to obtain the most well-known fixed point theorems in analytical fixed point theory. Actually, our results include the historically well-known theorems due to Brouwer, Schauder, Tychonoff, Kakutani, Hukuhara, Bohnenblust and Karlin, Fan, Glicksberg, Fort, Himmelberg, Granas and Liu, Lassonde, Smart, Chang and Yen, and others”.

MSC:
47H10 Fixed-point theorems
47H04 Set-valued operators
54C60 Set-valued maps in general topology
54H25 Fixed-point and coincidence theorems (topological aspects)
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