Asymptotics for Solutions of Linear Differential Equations Having Turning Points with Applications

Asymptotics are built for the solutions $y_j(x,\lambda)$, $y_j^{(k)}(0,\lambda)=\delta_{j\,n-k}$, $0\le j,k+1\le n$ of the equation $L(y)=\lambda p(x)y,\quad x\in [0,1],$ where $L(y)$ is a linear differential operator of whatever order $n\ge 2$ and $p(x)$ is assumed to possess a finite number of turning points. The established asymptotics are afterwards applied to the study of: the existence of infinite eigenvalue sequences for various multipoint boundary problems posed on $L(y)=\lambda p(x)y,\quad x\in [0,1],$, especially as $n=2$ and $n=3$ (let us be aware that the same method can be successfully applied on many occasions in case $n>3$ too), and asymptotical distribution of the corresponding eigenvalue sequences on the complex plane.