Consider some function $f:\mathbb{R}\times X \times \Omega,$ $\Omega \subset X$ and the differential equation
\begin{equation} \dot{x}(t)=f(t,x,p), \end{equation}
where $p \in \Omega$ is some parameter. Then, if $f$ is Lipschitz-continuous in $p$ we know that also $x$ is Lipschitz-continuous in $p$.
Now, what I try to see is some generalization. Given some sequence of $f_n$ such that $\| f_n(t,x,\cdot) \|_{C^{0,1}(\Omega)}\leq c$ with $c$ independent of $n$ I would like to have $\| x_n(t,\cdot) \|_{C^{0,1}(\Omega)}\leq c'$ for the corresponding solutions $x_n$. Unfortunately, I don't see how one could prove this. I assume, that the above statement is proved by showing
\begin{equation} \|x(t,\cdot)\|_{C^{0,1}(\Omega)} \leq c \| f(t,x,\cdot) \|_{C^{0,1}(\Omega)} \end{equation}
for some further constant $c$, if this is the case my desired estimate would of course follow immediately. But I can't find a proof for the above statement, so I don't know if it really works like this. I appreciate any hint about how to prove any of the two statements. Thank you!