For each value of a parameter $a \in \mathbb{R}$, let $x(a,t)$ be defined by the ODE


where $F$ is (say) smooth and, for each fixed $a$, Lipschitz in $x$. It is well known in this case that $x$ is well-defined and is smooth with respect to $a$.

Question (general): What is known about the derivatives of $x$ with respect to $a$?

If this question is too general to be helpful, here's a more specific one:

Question (specific): If $F(a,x,t)$ is zero unless $x \in [-B,B]$ for some constant $B$, does this imply that all the derivatives of $x$ with respect to $a$ are bounded functions of $t$?

(An answer to either question could be a reference to a book)

The obvious thing to do is to write down an ODE satisfied by $y:=\frac{\partial x}{\partial a}$, which is, I think,

$$\frac{dy}{dt}=\frac{\partial F}{\partial a}+\frac{\partial F}{\partial x}y$$

and this could be used to study the first derivative (and similar for the higher ones). But these equations get a little messy for higher derivatives.



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