Applications of Michael's selection theorems to fixed point theory (Q924279)

Applying some of Michael's selection theorems, from recent fixed point theorems on upper semicontinuous multimaps, the author deduces generalizations of the classical Bolzano intermediate value theorem, several fixed point theorems on multimaps defined on almost convex sets, almost fixed point theorems, coincidence theorems, and collectively fixed point theorems. These results are related mainly to Michael maps, that is, lower semicontinuous multimaps having nonempty closed convex values. In Section 2, three principal selection theorems of \textit{E. Michael} are introduced [Ann. Math. (2) 63, 361--382 (1956; Zbl 0071.15902); Proc. Am. Math. Soc. 17, 1404--1406 (1966; Zbl 0178.25902); Fundam. Math. 111, 1--10 (1981; Zbl 0455.54012)]. One of them is applied to a generalization of Bolzano's theorem for Michael maps. Section 3 deals with a unified fixed point theorem on convex-valued upper hemicontinuous multimaps due to the author [J. Korean Math. Soc. 29, No. 1, 191--208 (1992; Zbl 0758.47048); Acta Math. Vietnam 27, No. 2, 141--150 (2002; Zbl 1026.54044)] and some of its consequences. These are applied to obtain another multi-valued version of Bolzano's theorem and a fixed point theorem for Michael maps. In Sections 4 and 5, the author deduces fixed point theorems for Michael maps defined on almost convex sets and some almost fixed point theorems. These new results generalize a number of known ones. Section 6 deals with existence theorems of coincidence points of multimaps in the class \(\mathfrak B\) of multimaps due to the author [J. Math. Anal. Appl. 329, No. 1, 690--702 (2007; Zbl 1117.54051)] with continuous functions or Michael maps. Finally in Section 7, the author obtains some collectively fixed point theorems for families of Michael maps, which would be applicable to equilibrium problems.