Fixed point theorems for weakly inward multivalued mappings and their randomizations (Q1329294)

The following theorem is the main result of the paper: Let \(D\) be a closed subset of a Banach space \(X\), and \(T\) a compact- valued multifunction from \(D\) to \(X\). If \(T\) is weakly inward and contractive then it has a fixed point. The proof is based on the Caristi fixed point theorem. Then the authors give a random analogue of the above result: If \(T\) depends measurably on a random parameter, then there exists a fixed point which also depends measurably on this parameter. \{Reviewer's remark: A more general result than Lemma 3.1 was already given by \textit{N. S. Papageorgiou} [J. Aust. Math. Soc., Ser. A 45, No. 2, 204-216 (1988; Zbl 0666.60058), Theorem 3.1]\}.