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, : be a proper, bounded below and lower semicontinuous function and multimap T: is such that for every , there exists satisfying
Then T has a fixed point.
It is shown that this result is equivalent to the -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)].