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Mizoguchi-Takahashi’s fixed point theorem is a real generalization of Nadler’s. (English) Zbl 1137.54026

The author gives a very simple proof of Banach’s contraction principle for set-valued mappings with nonempty bounded closed values in a complete metric space due to N. Mizoguchi and W. Takahashi [J. Math. Anal. Appl. 141, No.1, 177–188 (1989; Zbl 0688.54028)]. It is illustrated by an example that this result is a real generalization of a fixed point theorem of S. B. Nadler Jr. [Pac. J. Math. 30, 475–488 (1969; Zbl 0187.45002)], in contrast to A. A. Eldred, J. Anuradha and P. Veeramani [J. Math. Anal. Appl. 336, No. 2, 751–757 (2007; Zbl 1128.47051)].

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
54C60 Set-valued maps in general topology
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References:

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