We have the following result for continuous dependence of the initial value for ODEs with a continuous right-hand side (Satz 8.18 in https://www.mathematik.hu-berlin.de/~baum/Skript/DGL-2012.pdf):
Let $F:U \subset \mathbb{R}^n \times \mathbb{R} \rightarrow \mathbb{R}^n$ on $U$ be continuous and Lipschitz continuous with respect to the $\mathbb{R}^n$-variable with Lipschitz constant $L$. Let $(x_0,t_0),(x_0^*,t_0) \in U$ and $\varphi_{x_0},\varphi_{x_0^*} : [t_0 - \epsilon, t_0 + \epsilon] \rightarrow \mathbb{R}^n$ be solutions of the ODE $x'=F(x,t)$ with initial values $\varphi_{x_0}(t_0)=x_0$ and $\varphi_{x_0^*}(t_0)=x_0^*$. Then: $$| \varphi_{x_0}(t)-\varphi_{x_0^*}(t) | \leq |x_0-x_0^*| \cdot e^{L|t-t_0|} \forall t \in [t_0-\epsilon,t_0+\epsilon].$$
Question: is there a similar result for Holder continuity? My dream result would be
Let $F:U \subset \mathbb{R}^n \times \mathbb{R} \rightarrow \mathbb{R}^n$ on $U$ be smooth and $\alpha$-Holder continuous with respect to the $\mathbb{R}^n$-variable, with $\alpha$-Holder norm bounded by $L$. Let $(x_0,t_0),(x_0^*,t_0) \in U$ and $\varphi_{x_0},\varphi_{x_0^*} : [t_0 - \epsilon, t_0 + \epsilon] \rightarrow \mathbb{R}^n$ be solutions of the ODE $x'=F(x,t)$ with initial values $\varphi_{x_0}(t_0)=x_0$ and $\varphi_{x_0^*}(t_0)=x_0^*$. Then there exist a universal constant $c$ independent of $F$, and $\beta \in (0,1)$ such that $$| \varphi_{x_0}(t)-\varphi_{x_0^*}(t) | \leq c|x_0-x_0^*|^{\beta} \cdot e^{L|t-t_0|} \forall t \in [t_0-\epsilon,t_0+\epsilon].$$
Note that I am happy to assume my function is smooth in order to have a unique solution. I also note that I can use a $C^k$-bound for $F$ to get a $C^k$ bound for the solution depending on the initial value. But what I need is a $C^{0,\beta}$-bound for the solution that only depends on the $C^{0,\alpha}$-bound for $F$. I can imagine something like this exists for $\beta=\alpha/2$, but maybe even $\beta=\alpha$ is possible.
I know that the proof of the above theorem cannot be adapted to prove my estimate. I also found "Agarwal, Lakshmikantham: Uniqueness and nonuniqueness criteria for ordinary differential equations" to make some statements about uniqueness of the solution for a Holder-continuous right-hand side $F$. But I did not find anything resembling the estimate I need.
Context: I have two metrics on a compact manifold, $g_1, g_2$, satisfying the estimate $||g_1-g_2||_{C^{1,\alpha},g_1}<c_1$. I also have a vector $\eta$ satisfying $|| \eta ||_{C^{1,\alpha},g_1} < c_2$. I would like to have an estimate $|| \eta ||_{C^{1,\beta},g_2} < F(c_1,c_2)$, where $\beta$ can depend on $\alpha$, but should not depend on $\eta$, and $F(c_1,c_2)$ is some universal expression in $c_1$ and $c_2$. I believe that my dream ODE result from above would give me such an estimate.
