Hartman–Wintner theorem

Remarks

The proof goes through the Riccati equation. A nonoscillatory equation has a solution $y > 0$ on some interval $[T, \infty)$, and $w = y'/y$ satisfies $w' + w^2 + q = 0$ there. Integrating gives $w(t) = w(T) - \int_T^t q - \int_T^t w^2$. Averaging this identity and using the lower bound on $A$, one shows that $\int_T^\infty w^2 < \infty$, and then the average $A(t)$ converges.

Read as a test for oscillation, the theorem says: if $\liminf A(t) > -\infty$ and either $\liminf A(t) < \limsup A(t)$ or $A(t) \to \infty$, then the equation is oscillatory. The case $A(t) \to \infty$ is Wintner's oscillation criterion.

Example: for $q = 1$ on $[0, \infty)$ one gets $A(t) = t/2 \to \infty$, so the equation is oscillatory; indeed $\sin t$ is a solution. For $q(t) = 1/(4t^2)$ on $[1, \infty)$ the equation has the positive solution $\sqrt{t}$, so it is nonoscillatory. Here $A(t) = (t - 1 - \ln t)/(4t)$, which tends to the finite limit $1/4$, as the theorem predicts.

Going deeper

For a student

The Sturm comparison theorem compares $q$ with a constant or an Euler coefficient pointwise. The Hartman–Wintner theorem uses only averages of $q$, so it also applies when $q$ changes sign. It improves Wintner's criterion, which asks that $A(t) \to \infty$. The key tool is the Riccati substitution $w = y'/y$, which turns a nonoscillatory linear equation into a first-order nonlinear one with a solution on a half-line. Later criteria of Kamenev and Philos replace the weight $1/t$ by other weights in the same Riccati argument.