Canonical forms and principal systems for general disconjugate equations

Abstract:

It is shown that the disconjugate equation (1) $Lx \equiv (1/{\beta _n})(d/dt) \cdot (1/{\beta _{n - 1}}) \cdots (d/dt)(1/{\beta _1})(d/dt)(x/{\beta _0}) = 0$ , a $< t < b$, where ${\beta _i} > 0$, and (2) ${\beta _i} \in C(a,b)$, can be written in essentially unique canonical forms so that ${\smallint ^b}{\beta _i}dt = \infty ({\smallint _a}{\beta _i}dt = \infty )$ for $1 \leq i \leq n - 1$. From this it follows easily that (1) has solutions ${x_1}, \ldots ,{x_n}$ which are positive in (a, b) near $b(a)$ and satisfy ${\lim _{t \to b}} - {x_i}(t)/{x_j}(t) = 0({\lim _{t \to a}} + {x_i}(t)/{x_j}(t) = \infty )$ for $1 \leq i < j \leq n$. Necessary and sufficient conditions are given for (1) to have solutions ${y_1}, \ldots ,{y_n}$ such that ${\lim _{t \to b}} - {y_i}(t)/{y_j}(t) = {\lim _{t \to a}} + {y_j}(t)/{y_i}(t) = 0$ for $1 \leq i < j \leq n$. Using different methods, P. Hartman, A. Yu. Levin and D. Willett have investigated the existence of fundamental systems for (1) with these properties under assumptions which imply the stronger condition $(2’){\beta _i} \in {C^{(n - i)}}(a,b)$.