Abstract
We establish the existence of a best proximity pair for which the best proximity set is nonempty for a finite family of multimaps whose product is either an
-multimap or a multimap
such that both
and
are closed and have the
KKM
property for each Kakutani multimap
. As applications, we obtain existence theorems of equilibrium pairs for free 1-person games as well as for free 1-person games. Our results extend and improve several well-known and recent results.