# Hölder continuous dependence on parameters for solutions of ODE

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 $$||g_1-g_2||_{C^{1,\alpha},g_1}. I also have a vector $$\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.

The quoted result is a Grönwall type estimate, $$|x(t)-x^*(t)|\le |x(t_0)-x^*(t_0)|+\int_{t_0}^tL·|x(s)-x^*(s)|\,ds$$ results in the exponential upper bound.
With $$α$$-Hölder continuity of $$F$$ in $$x$$-direction one gets $$|x(t)-x^*(t)|\le |x(t_0)-x^*(t_0)|+\int_{t_0}^tL·|x(s)-x^*(s)|^α\,ds$$ with $$α\in(0,1)$$. Here the equation for the upper bound is $$u'(t)=Lu(t)^α$$, which has the solution $$u(t)^{1-α}-u(t_0)^{1-α}=L(1-α)(t-t_0) \implies u(t)=\left(u(t_0)^{1-α}+L(1-α)(t-t_0)\right)^{\frac1{1-α}}$$ which implies for the Grönwall result $$|x(t)-x^*(t)|\le\left(|x(t_0)-x^*(t_0)|^{1-α}+L(1-α)|t-t_0|\right)^{\frac1{1-α}}$$