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On a fixed point problem of Reich. (English) Zbl 0874.47027
Let \(X\) be a complete metric space, \(CB(X)\) the hyperspace of all closed bounded \(M\subseteq X\) with the Hausdorff metric, and \(F:X\to CB(X)\) a certain local contraction. If \(F\) is compact-valued, then S. Reich [Boll. Unione Mat. Ital., IV. Ser. 5, 26-42 (1972; Zbl 0249.54026)] has shown that \(F\) has a fixed point. Here, the author proves the same for \(F\) just closed-valued.

47H06 Nonlinear accretive operators, dissipative operators, etc.
47H10 Fixed-point theorems
54H25 Fixed-point and coincidence theorems (topological aspects)
Full Text: DOI
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[6] Simeon Reich, Some problems and results in fixed point theory, Topological methods in nonlinear functional analysis (Toronto, Ont., 1982) Contemp. Math., vol. 21, Amer. Math. Soc., Providence, RI, 1983, pp. 179 – 187. · Zbl 0531.47048
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