The sufficiency of the Matkowsky condition in the problem of resonance

Abstract:

We consider the sufficiency of the Matkowsky condition concerning the differential equation $\varepsilon y'' + f(x,\varepsilon )y’ + g(x,\varepsilon )y = 0\;( - a \leqslant x \leqslant b)$ under the assumption that $f(0,\varepsilon ) = 0$ identically in $\varepsilon ,{f_x}(0,\varepsilon ) \ne 0$ with $f > 0$ for $x < 0$ and $f < 0$ for $x > 0$. Y. Sibuya proved that the Matkowsky condition implies resonance in the sense of N. Kopell if $f$ and $g$ are convergent power series for $|\varepsilon | < \rho \;(\rho > 0),f(x,0)=-2x$ and the interval $[ - a,b]$ is contained in a disc $D$ with center at $0$. The main problem in this work is to remove from Sibuya’s result the assumption that $D$ is a disc.