On Boundary Layer Problems Exhibiting Resonance

We consider singularly perturbed boundary value problems for differential equations of the form $\varepsilon y'' + f(x;\varepsilon )y' + g(x,\varepsilon )y = h(x,\varepsilon )$, where f changes sign at one or more points in the interval under consideration. Such points are sometimes referred to as turning points. We show that various anomalies can occur in such problems. One of these anomolies is the phenomenon of resonance. We present criteria for resonance to occur, and derive uniformly valid asymptotic expansions for the solution of such problems.