We establish the existence of an arbitrary number of positive solutions to
the 2mth order Sturm-Liouville type problem
(-1)(m)((2m))(y) (t) = f(t,y(t)), 0 less than or equal to t less than or eq
ual to 1, alphay((2i))(0) - betay((2i+1))(0) = 0, 0 less than or equal to i
less than or equal to m - 1, gammay((2i))(1) + deltay((2i+1))(1) = 0, 0 le
ss than or equal to i less than or equal to m - 1,
where f : [0, I] x [0, infinity) --> [0, infinity) is continuous. We accomp
lish this by making growth assumptions on f which we state in terms which g
eneralize assumptions in recent works regarding superlinear and/or sublinea
r growth in f.