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Can I use Sturm's comparison theorem to say something about the average?

Let $f(t)$ be a continuous function such that $\lim_{t \to \infty } f(t) = \infty$. Let us consider the following function

$$ \ddot{x}(t) + t^2 x(t) = 0. $$

Let $C$ be a constant. Then, we can find $T$ such that $C^2 < t^2$ for all $t > T$. By Sturm's comparison theorem, exist $t^*$ in $(\frac{2\pi k}{C},\frac{2\pi (k+1)}{C})$ such that $x(t^*) = 0$ for every $k$ big enough.

Can we concluded the following?

$$ \lim_{t \to \infty} x(t) =0 $$

or

$$ \lim_{t \to \infty} \frac{1}{t}\int_0^t s x(s) ds =0 $$

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  • $\begingroup$ your integration is wrong. $\endgroup$
    – alejandro
    Feb 26 at 13:26
  • $\begingroup$ Right, this was nonsense. $\endgroup$
    – Gerd
    Feb 26 at 21:37

1 Answer 1

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Sturm's comparison theorem is not the correct approach. If we set up $x(t) = \sqrt{x}u(\frac{x^2}{2})$, then $u$ satisfies Bessel equation. Using the asymptotic decay of Bessel function we can conclude the desired results.

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