I have a vectorial, non-linear second order ordinary differential equation $$Z''=f(Z)$$ for which I have a solution $Z^0$ on $[0,1]$ with $Z^0_i(0)=0$ and $Z^0_i(1)=1$. I would like to know under which kind of conditions on $f$ it is true that no other solution with same endpoint values can exist. The dimension $1$ case would already be interesting for me to understand, even if I am in fact truly interested in systems.

I am pretty sure this kind of question is very classical, but I have some trouble finding relevant keywords to make my way in the literature. Any pointer would be appreciated.

By the way, my initial motivation is a Riemannian geometry problem, but I do not think it is especially relevant to this question.