In this paper we examine nonlinear two point boundary value problems, with Sturm-Liouville type boundary conditions, in a Hilbert Space. Our technique involves using the Topological Transversality Theorem of A. Granas which relies on the notions of an essential map and a priori bounds on solutions. We obtain existence results for a wide class of problems with our nonlinear term satisfying a Bernstein-Nagumo growth condition.