This topic has been inspired by some time spending on trying to refining my knowledge about PDE's in general. Then everybody knows how to solve $$y''(t)=\pm y(t).$$ Then I tried to slightly modify the question and I focused on $$(\therefore)\;y''(t)=f(y(t)),\; f\in C^1(\mathbb R,\mathbb R).$$ What I was trying to do was to derive some general properties about the solutions to this equation. In particular I ask to you, since I was not able to answer myself:
Must a solution of $(\therefore)$, not identically zero, have necessarily a finite number of zeroes on $[0,1]$? My idea was to derive an estimate like
$$|y(\eta)-y(\xi)|\leq C|\eta-\xi|^p,\; p>1;$$ Moreover, if a solution $y$ were to have an infinite number of zeroes in $[0,1]$, the the set of zeroes should have an accumulation point, and in this point all the derivatives should be equal to $0$ by continuity, then maybe the function should remain to much squeezed to be different from zero.
Hope you can help me because this interests me a lot.
Many thanks for your attention.
(I tried to post this on mathlinks as well but nobody answered me yet)