Z be a multifunction having nonempty convex images and open fibers. Then T has a continuous selection. For this purpose we introduce convex structures on topological spaces ones of more general than those of topological vector spaces, or topological convexity structures due to Michael, Van de Vel, Horvath, and others. We are able to construct a convexity structure for a wide class of topological spaces, which makes it possible to prove a generalization of the following purely topological fixed point theorem. Eilenberg-Montgomery Fixed Point Theorem.
X be an upper semicontinuous multifunction with nonempty closed acyclic values.
Then F has a fixed point. It is generalized in a different direction in A Lefschetz-type coincidence theorem . Dissipativity in the plane Russian Academy of Sciences. The method consists in the construction of a system of compact sets covering the plane, whose boundaries are given by trajectories of a certain auxiliary system, and the trajectories of the given system intersect them from outside in. We extend this result to include nonautonomous spaces of solutions due to V. V. Filippov.