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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

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