The authors prove existence theorems for invariant best approximation of compatible maps which are based on common fixed point theorems for noncommuting maps. The basic result can be described as follows: Let
be a nonempty subset in a metric space
maps such that
. Assume that
is complete and contained in
and that there exists a
. Then there is a point which is the unique coincidence point of
and this point is the unique common fixed point of both maps.