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Fixed point theorems for multivalued mappings on complete metric spaces. (English) Zbl 0688.54028
The authors give the following “multi-version” of Caristi’s fixed point theorem [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 \[ \psi (y)+d(x,y)\leq \psi (x). \] Then T has a fixed point.
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.
\(\{\) Reviewer’s remark: Another generalization of the Caristi’s theorem on multifunctions was given in the work of J. Madhusudana Rao [Bull. Math. Soc. Sci. Math. Répub. Soc. Roum., Nouv. Ser. 29(77), No.1, 79-80 (1985; Zbl 0561.54041)]\(\}\).
Reviewer: V.V.Obukhovskij

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