Focal points for a linear differential equation whose coefficients are of constant signs

Abstract:

The differential equation considered is ${y^{(n)}} + \Sigma {{p_i}(x){y^{(i)}}} = 0$, where ${\sigma _i}{p_i}(x) \geqslant 0,i = 0,\ldots ,n - 1,{\sigma _i} = \pm 1$. The focal point $\zeta (a)$ is defined as the least value of s, $s > a$, such that there exists a nontrivial solution y which satisfies ${y^{(i)}}(a) = 0,{\sigma _i}{\sigma _{i + 1}} > 0$ and ${y^{(i)}}(s) = 0$, ${\sigma _i}{\sigma _{i + 1}} < 0$. Our method is based on a characterization of $\zeta (a)$ by solutions which satisfy ${\sigma _i}{y^{(i)}} > 0,i = 0,\ldots ,n - 1$, on $[a,b]$, $b < \zeta (a)$. We study the behavior of the function $\zeta$ and the dependence of $\zeta (a)$ on ${p_0},\ldots ,{p_{n - 1}}$ when at least a certain ${p_i}(x)$ does not vanish identically near a or near $\zeta (a)$. As an application we prove the existence of an eigenvalue of a related boundary value problem.