The Coincidence Problem for Compositions of Set-valued Maps

If A is compact, then A and B have a coincidence, that is, there exists (z0, Vo) € X x Y with y0 G A(x0) D B(x0). [Recall that a nonempty compact space is acyclic if all its reduced Cech homology groups over Q vanish.] The proof of Theorem 0 relies on a Lefschetz-type fixed point theorem of Gorniewicz and Granas [14] which is itself based on sophisticated homological machinery. One of our concerns was to provide a simple proof of the convex case, that is when A is use and has non-empty convex compact values (such a map will be called a K-map). In doing so, we were driven to give an elementary proof of a fairly general fixed point theorem for compositions of K-maps defined on a general extension space containing the locally convex and the not necessarily locally convex cases. Weaker results were recently and independently obtained by Lassonde [24]. However, our fixed point theorems are different from Lassonde's in the sense that the spaces are more general and an approximate selection technique is used here rather then the approximation by simplicial maps (see Kakutani [21], Ha [18], Lassonde [24]). We are thus able to deduce some general coincidence theorems for compositions of set-valued maps.