Coincidences and fixed points of multifunctions into trees
Helga Schirmer · Pacific Journal of Mathematics · 1970
The main purpose of this paper is to find conditions on an upper semi-continuous (use) multifunction φ from a compact Hausdorff space X onto a tree T so that it has a coincidence with any multifunction f:X->T which is either continuous or use and connected-valued.It is shown that it is sufficient (but not necessary) that φ be either open or monotone.This result contains as special cases known conditions for coincidence producing single-valued maps onto trees as well as known fixed point theorems for multif unctions on trees.It is used to obtain a new result on fixed points, namely that any composite of an use and connected-valued and a continuous multifunction of a tree into itself has a fixed point.All proofs make use of the order-theoretic characterization of trees by L.