The main purpose of this paper
is to find conditions on an upper semi-continuous (usc) multifunction φ from
a compact Hausdorff space X onto a tree T so that it has a coincidence
with any multifunction ψ : X → T which is either continuous or usc 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 multifunctions on trees. It is used to obtain a new result on fixed
points, namely that any composite of an usc 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. E. Ward,
Jr.
