We obtain sharp
differentiability results for subfunctions for second order ordinary differential
equations y′′= f(x,y,y′) on [a,b]. In the process we show that a subfunction satisfies
a second order differential inequality similar to that satisfied by a lower solution. We
show that a subfunction can be used in maximum principle arguments in the
same way one uses a lower solution. As an application of these results we
give necessary and sufficient conditions on a function in order that there
is a differential equation for which it is a subfunction. We use our results
together with the Perron method to improve on some existence results for
two point boundary value problems obtained by Jackson, using Perron’s
method.
