We establish existence results
for two point boundary value problems for second order ordinary differential
equations of the form y′′= f(x,y,y′), x ∈ [0,1], where f is continuous and there
exist lower and upper solutions. First we consider boundary conditions of
the form G((y(0),y(1));(y′(0),y′(1))) = 0, where G is continuous and fully
nonlinear. We introduce compatibility conditions between G and the lower
and upper solutions. Assuming these compatibility conditions hold and, in
addition, f satisfies assumptions guarenteeing a’priori bounds on the derivatives
of solutions we show that solutions exist. In the case the lower and upper
solutions are constants one of our results is closely related to a result of
Gaines and Mawhin. Secondly we consider boundary conditions of the form
(y(i),y′(i)) ∈𝒥 (i), i = 0,1 where the 𝒥 (i) are closed connected subsets
of the plane. We introduce various compatibility type conditions relating
the 𝒥 (i) and the lower and upper solutions and show each is sufficient to
construct a compatible G which defines these boundary conditions. Thus
our existence results apply. Almost all the standard boundary conditions
considered in the literature assuming upper and lower solutions are, or can be,
defined by compatible G and their associated existence results follow from
ours; in many cases we can improve these results by deleting some of their
assumptions.
