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# Fixed points and selections of set valued maps on spaces with convexity by Saveliev

*Fixed points and selections of set valued maps on spaces with convexity* by Peter Saveliev

International Journal of Mathematics and Mathematical Sciences, 24 (2000) 9, 595-612. Also a talk at at the Joint Mathematics Meeting in January 2000. Reviews: MR 2001h:47097, ZM 0968.47016.

We provide two results that unite the following two pairs of theorems respectively.

First:

- Kakutani fixed point theorem. Let $X$ be a nonempty convex compact subset of a locally convex Hausdorff topological vector space, and let $F:X \rightarrow Y$ be an upper semicontinuous multifunction with nonempty closed convex values. Then $F$ has a fixed point.
- Browder fixed point theorem. Let $X$ be a nonempty convex compact subset of a Hausdorff topological vector space, and let $F:X\rightarrow X$ be a multifunction with nonempty convex images and fibers relatively open in $X$. Then $F$ has a fixed point.

Second:

- Michael selection theorem. Let $X$ be a paracompact Hausdorff topological space, and let $Y$ be a Banach space. Let $T:X \rightarrow Y$ be a lower semicontinuous multifunction with nonempty closed convex images. Then $T$ has a continuous selection, i.e. a map $g:X\rightarrow Y$ such that $g(x)\in T(x)$ for all $x$.
- Browder selection theorem. Let $X$ be a paracompact Hausdorff topological space, and let $Z$ be any topological vector space. Let $T:X\rightarrow Z$ 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 that are 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. Let $X$ be an acyclic compact ANR, and let $F:X\rightarrow X$ be an upper semicontinuous multifunction with nonempty closed acyclic values. Then $F$ has a fixed point.

This theorem is especially important as it is used in proving the existence of periodic solutions of differential inclusions (multivalued differential equations), see Dissipativity in the plane and The dissipativity of the generalized Lienard equation. It is generalized in a different direction in A Lefschetz-type coincidence theorem by Saveliev.

This fixed point theorem is just one of the scores generated by Problem 54 of The Scottish Book. However I believe that I don't just add another one to the list but instead reduce the total number. Selection theorems are just as numerous, see "Continuous selections of multivalued mappings" by D. Repovs and P.V. Semenov.

Full text: Fixed points and selections of set valued maps on spaces with convexity (17 pages)