Fixed points of subopen multifunctions

J. F. McClendon
1982 Proceedings of the American Mathematical Society  
A fixed point theorem of the Lefschetz type is proved for subopen (e.g., open-graph) multifunctions. A coincidence version is also given. The fixed point theorem of [5, 6] is generalized here to fixed point theorems of the Lefschetz type. The following will be proved (see § 1 for terminology). 3.2 Theorem. Let X be a compact finite-dimensional ANR and m: X -^> X a subopen multifunction with infinitely connected values. Then the Lefschetz number L(m) = "Z™=0(-l)qtr(mq) is defined and if L(m) =£
more » ... ned and if L(m) =£ 0 then m has a fixed point. Corollary. Let X be an acyclic compact finite-dimensional ANR and m: X -> X a subopen multifunction with infinitely connected values. Then m has a fixed point. The corollary includes the results of [5, 6], mentioned above. In §4 a coincidence form of these is given.
doi:10.1090/s0002-9939-1982-0640246-x fatcat:2pshvd2u3vbcfai4qpkfpvyxwy