Usbrth

Consider the ODE $x'=f(x,t)$ in $mathbbR^n$ and $tinmathbbR$. Let $gamma(t)$ a soluitino for the ODE such that $gamma(0)=0$ and $gamma(1)=x_0$. So, given a neighborhood $U_x_0$ of $x_0$, there is $epsilon>0$ such that, if $g(x,t)$ satisfies $|g(x,t)-f(x,t)|<epsilon$ for all $x$ and $t$, so the solution $eta(t)$ of $x'=g(x,t)$ such that $eta(0)=0$ satisfies $eta(1)in U_x_0$.

Is this statement true or false? Intuitively, seems true, but I don't know how to demonstrate.

** When $f(t)$ and $g(t)$ do not depend on $x$ this works:

$|doteta(t) - dot gamma(t)| = |f(t)-g(t)|<epsilon$, so integrate $-epsilon < dot eta(t) - dot gamma(t) < epsilon$

and use that $eta(0) = gamma(0) = 0$ to get:

$|eta(t) - gamma(t) | < epsilon t$.

Then set $t = 1$ to get that $eta(1)$ is $epsilon$-close to $x_0 = gamma(1)$.

So, for a general neighborhood of $x_0$, it will depend on $t$ (how far you want to send the solutions) and how good your bound $epsilon$ for $|f - g|$ is. In the worst case, an $O(epsilon)$ perturbation can have an $O(1)$ effect on solutions within time scale $O(1/epsilon)$ (but of course for certain examples things could be better)

**edit: thanks to LutzL and user539887's comments, we need to add conditions on how to compare $f(x,t)$ and $g(y,t)$ when $xne y$. For example, when $f$ has a Lipschitz condition $|f(x,t) - f(y,t)|le M|x-y|$, here is a way we can work out some bounds:

Set $u(t) = eta(t) - gamma(t)$, then $u(0) = 0$ and

$|dot u| = |g(eta(t), t) - f(gamma(t), t) + f(eta(t), t) - f(eta(t), t)|<epsilon + M |u|$.

then compare directly with solutions of $dot u_+ = epsilon + M|u_+|, dot u_- = -epsilon - M|u_-|$ to get

$-fracepsilonM(e^Mt - 1) = u_-(t) le u(t) le u_+(t) = fracepsilonM(e^Mt - 1)$ for $tge 0$

Or $|u(t)|le fracepsilonM(e^Mt-1)$ for $tge 0$.