Is there a multi-valued fixed point theorem for complex projective Hilbert spaces, or equivalently, Kahler manifolds with positive curvature?
In other words, an analogue of Kakutani's theorem for omplex projective Hilbert spaces. I am aware of Frankel's theorem (https://msp.org/pjm/1961/11-1/pjm-v11-n1-p12-p.pdf) (theorem 4) which assures a fixed point for holomorphic functions from such a space to itself. Is anyone aware of work on extending Frankel's theorem to the multivalued case for manifolds? Thanks in advance.