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Zbl 0688.54028
Mizoguchi, Noriko; Takahashi, Wataru
Fixed point theorems for multivalued mappings on complete metric spaces. (English)
[J] J. Math. Anal. Appl. 141, No.1, 177-188 (1989). ISSN 0022-247X

The authors give the following ``multi-version'' of Caristi's fixed point theorem [{\it J. Caristi}, Trans. Am. Math. Soc. 215, 241-251 (1976; Zbl 0305.47029)]. Let (X,d) be a complete metric space, $\psi$ : $X\to (- \infty,+\infty]$ be a proper, bounded below and lower semicontinuous function and multimap T: $X\to P(X)$ is such that for every $x\in X$, there exists $y\in Tx$ satisfying $$ \psi (y)+d(x,y)\le \psi (x). $$ Then T has a fixed point. \par It is shown that this result is equivalent to the $\epsilon$-variational principle of Ekeland. Then it is used to generalize Nadler's fixed point theorem and to obtain a common fixed point theorem for a single-valued map and a multimap. Next, some generalizations of Reich's fixed point theorems for multimaps of contractive type are considered. \par $\{$ Reviewer's remark: Another generalization of the Caristi's theorem on multifunctions was given in the work of {\it J. Madhusudana Rao} [Bull. Math. Soc. Sci. Math. Répub. Soc. Roum., Nouv. Ser. 29(77), No.1, 79-80 (1985; Zbl 0561.54041)]$\}$.
[V.V.Obukhovskij]

MSC 2000:: *54H25 Fixed-point theorems in topological spaces
54C60 Set-valued maps

Keywords: complete metric space; $\epsilon $ -variational principle; common fixed point; contractive type

Citations: Zbl 0305.47029; Zbl 0561.54041

Cited in: Zbl pre06151096 Zbl 1235.54040 Zbl 1223.54059 Zbl 1206.54050 Zbl 1198.65098 Zbl 1179.54053 Zbl 1165.54306 Zbl 1213.54063 Zbl 1166.54307 Zbl 1137.54026 Zbl 1094.47049 Zbl 1081.47069 Zbl 1054.47036 Zbl 0840.47041 Zbl 0835.54028 Zbl 0840.54032 Zbl 0768.54033 Zbl 0768.54032

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