Quantitative estimate on continuity with respect to parameter of ODE

Question

Let $V:\mathbb{R} \rightarrow \mathbb{R}$ be a bounded, $\text{Holder}$ continuousfunction with degree $\delta\in (0,1]$ such that $\inf_{\mathbb{R}} V = 0$ but $V(x) > 0$ for all $x\in \mathbb{R}$. Let $r \geq 0$ be the parameter, we consider $\eta_r(s)$ be the solution to the following ODE: $$ \begin{cases} \dot{\eta_r}(s) &= \sqrt{r+V(\eta(s))}, \qquad{s>0},\\ \eta(0) &=0. \end{cases}$$ I am concerning about the following question, as $r\rightarrow 0$, it is reasonable to expect $\eta_r \rightarrow \eta_0$ in somesense. Is there anyway we can quantify the difference of $\eta_r(s)$ and $\eta_0(s)$ in terms of $r$ and $s$, which guarantee that the two solutions are close to each other provided $r$ very close to 0?

Answer 1

I only know how to do it for lipschitz functions. Also my estimate only gives locally uniformly convergence, but not the strong estimate you are looking for.

Assume that $V$ is lipschitz.

Fix $s_0$ and say $0\leq s \leq s_0$. Then by assumption $c_0:= \min_{t\in [0;s_0]} V(\eta_0(s)) >0$ and we obtain for $r\geq 0$ $$ \vert \eta_r(s) - \eta_0(s) \vert = \left\vert \int_0^s (\eta_r'(t)- \eta_0'(t)) dt \right\vert = \left\vert \int_0^s \left(\sqrt{r+ V(\eta_r(t))} - \sqrt{ V(\eta_0(t)} \right) dt \right\vert = \left\vert \int_0^s \frac{r + V(\eta_r(t))- V(\eta_0(t))}{\sqrt{r+V(\eta_r(t))} + \sqrt{V(\eta_0(t))}} \right\vert.$$ We also know that $V$ is Lipschitz continuous, and hence we get $$ \vert V(\eta_r(t))- V(\eta_0(t)) \vert \leq C \vert \eta_r(t) - \eta_0(t) \vert $$ This yields $$ \vert \eta_r(s) - \eta_0(s) \vert \leq \frac{rs}{\sqrt{c_0}} + \frac{C}{\sqrt{c_0}} \int_0^s \vert \eta_r(t) - \eta_0(t) \vert dt $$ Thus, we get from Gronwall's inequality $$\vert \eta_r(s) - \eta_0(s) \vert \leq \frac{rs}{\sqrt{c_0}} e^{s\frac{C}{\sqrt{c_0}}}.$$