Ackerberg-O'Malley resonance in boundary value problems with a turning point of any order

We consider an Ackerberg-O'Malley singular perturbation problem $\epsilon y'' + f(x,\epsilon)y' + g(x,\epsilon)y=0, y(a)=A, y(b)=B$ with a single turning point and study the nature of resonant solutions $y=\varphi(x,\epsilon)$, i.e. solutions for which $\varphi(x,\epsilon)$ tends to a nontrivial solution of $f(x,0)y'+ g(x,0)y=0$ as $\epsilon\to 0$. Many techniques have been applied to the study of this problem (WKBJ, invariant manifolds, asymptotic methods, spectral methods, variational techniques) and they have been successful in characterizing these resonant solutions when $f(x,0)$ has a simple zero at the origin. When the order of zero is higher the increase in complexity of the problem is significant. The existence of a nonzero formal power series solution is no longer necessary for resonance and resonant solutions are in general not smooth at the origin. We apply the method of blow up to study the nature of resonant solutions in this setting, using techniques from invariant manifold theory and planar singular perturbation theory. The main result is the sufficiency of the Matkowsky condition for turning points of arbitrary order (based on Gevrey-asymptotics), but we also give a characterization of the location of the boundary layer in resonant solutions.