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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.

##### MSC:
 47H06 Nonlinear accretive operators, dissipative operators, etc. 47H10 Fixed-point theorems 54H25 Fixed-point and coincidence theorems (topological aspects)
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##### References:
 [1] Thakyin Hu, Fixed point theorems for multivalued mappings, Canad. Math. Bull. 23 (1980), no. 2, 193 – 197. · Zbl 0436.54037 [2] Hwei Mei Ko and Yüeh Hsia Ts’ai, Fixed point theorems with localized property, Tamkang J. Math. 8 (1977), no. 1, 81 – 85. · Zbl 0361.54029 [3] Sam B. Nadler Jr., Some results on multi-valued contraction mappings, Set-Valued Mappings, Selections and Topological Properties of 2^{\?} (Proc. Conf., SUNY, Buffalo, N.Y., 1969) Lecture Notes in Mathematics, Vol. 171, Springer, Berlin, 1970, pp. 64 – 69. [4] Simeon Reich, Some fixed point problems, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 57 (1974), no. 3-4, 194 – 198 (1975) (English, with Italian summary). · Zbl 0329.47019 [5] Simeon Reich, Fixed points of contractive functions, Boll. Un. Mat. Ital. (4) 5 (1972), 26 – 42 (English, with Italian summary). · Zbl 0249.54026 [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 [7] Simeon Reich, A fixed point theorem for locally contractive multi-valued functions, Rev. Roumaine Math. Pures Appl. 17 (1972), 569 – 572. · Zbl 0239.54033
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