On the growth of solutions in the oscillatory case

Abstract:

Suppose that $A$ is a bounded continuously differentiable function from $[0,\infty )$ to the real $n \times n$ Hermitian matrices such that, for every $\varepsilon > 0$ and every $\lambda > 0$, there is an $a$ (depending on $\varepsilon$ and $\lambda$) such that ${D^2} - A - \varepsilon E$ and ${D^2} + A’/\lambda - \varepsilon E$ are disconjugate on $[a,\infty )$, where $E$ is the $n \times n$ identity matrix. It follows from the result of this paper that no solution of $({D^2} + A)f = 0$ can either grow or decay exponentially.