  We show that a continuous additive positively homogeneous map from a closed
not necessarily proper cone in a Banach space onto a Banach space is an open
map precisely when it is surjective. This generalization of the usual Open
Mapping Theorem for Banach spaces is then combined with Michael's Selection
Theorem to yield the existence of a continuous bounded positively homogeneous
right inverse of such a surjective map; a strong version of the usual Open
Mapping Theorem is then a special case. As another consequence, an improved
version of the analogue of And\^o's Theorem for an ordered Banach space is
obtained for a Banach space that is, more generally than in And\^o's Theorem, a
sum of possibly uncountably many closed not necessarily proper cones.
Applications are given for a (pre)-ordered Banach space and for various spaces
of continuous functions taking values in such a Banach space or, more
generally, taking values in an arbitrary Banach space that is a finite sum of
closed not necessarily proper cones.
