Geometric Theory of Differential Equations. I: Second Order Linear Equations

Abstract

In the first part of the paper, we develop a geometric theory of the equation $x'' + px = 0$ for p a generalized derivative and, simultaneously, the SL(2) differential geometry of curves with countably many cusps. After a discussion of Borůvka’s dispersion function for generalized p we use the dispersion function to obtain majorizations of the Lyapunov integral $(b - a)\int _a^b p\,dt$ on intervals of disconjugacy. As an application we obtain that the lower endpoint $\lambda _1 $ of the first interval of instability of a Hill equation $x'' +\lambda px = 0$ and period T is bounded above by $\pi ^2 (T\int _0^T p\,dt)^{ - 1} $. No other eigenvalue of the stability problem admits a similar upper bound.