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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 [{\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. 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 {\it 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

54H25Fixed-point and coincidence theorems in topological spaces
54C60Set-valued maps (general topology)
Full Text: DOI
[1] Assad, N. A.; Kirk, W. A.: Fixed point theorems for set-valued mappings of contractive type. Pacific J. Math. 43, 553-562 (1972) · Zbl 0239.54032
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