Sufficient Conditions for Ackerberg–O’Malley Resonance

Abstract

An investigation is made of the asymptotic nature of the solution of the boundary-value problem \[\varepsilon y'' + 2xA(\varepsilon ,x)y' - A(\varepsilon ,x)B(\varepsilon ,x)y = 0;\quad y(a) = l,\quad y(b) = m,\] as $\varepsilon \to 0$, where $A(\varepsilon ,x)$ and $B(\varepsilon ,x)$ are continuous real functions of $\varepsilon $ and x, $a < 0$, $b > 0$, and $A(\varepsilon ,x)$ is nonzero in $[a,b]$. Particular attention is paid to the problem of resonance, which arises when the limiting form of the solution exhibits an unusual lack of decay (in the case $A(\varepsilon ,x) < 0$), or an unusual rate of growth (in the case $A(\varepsilon ,x) > 0$). By application of a recent theory of differential equations with coalescing turning points sufficient conditions for resonance are established, both with and without the assumption that $A(\varepsilon ,x)$ and $B(\varepsilon ,x)$ are analytic functions of $\varepsilon $ and x. Illustrative examples are also included.