Are there theorems that could be applied to show the existence/uniqueness of solution of such system :
$$ (S) \left \{ \begin{aligned} p''&= f_1(b) \\ q'' &= f_2(x,p,q,b)\\ q~~ &= f_3(x,p,b) \end{aligned} \right. $$
where $f_1$, $f_2$ and $f_3$ are non linear functions ? and $p$, $b$ and $q$ are real and positive functions defined on a compact $[a,b] \subset \mathbb{R+}$ ?