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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, $\psi$ : $X\to \left(-\infty ,+\infty \right]$ be a proper, bounded below and lower semicontinuous function and multimap T: $X\to P\left(X\right)$ is such that for every $x\in X$, there exists $y\in Tx$ satisfying

$\psi \left(y\right)+d\left(x,y\right)\le \psi \left(x\right)·$

Then T has a fixed point.

It is shown that this result is equivalent to the $ϵ$-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.

$\left\{$ 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)]$\right\}$.

Reviewer: V.V.Obukhovskij

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