Generalization of Comparison Theorem

[Only a Mirage]

I was wondering if there is a generalization of the following (roughly stated) theorem to n-dimensional systems:

Given some restrictions on the functions $f$ and $g \in \Re$, let $y_s(t)$ and $x_s(t) \in \Re$ be solutions to the initial value problems:


$$\dot{x}(t) = f(x,t), x(t_0) = x_0\\ \dot{y}(t) = g(x,t), y(t_0) = y_0$$

Then if $f(x,t) > g(x,t)$ in some domain $D$ containing $x_0$ and $y_0$ and $x_0 \geq y_0$, $x_s(t) > y_s(t) \forall t > t_0$ when the trajectories stay in this domain $D$.

This is a rough statement of the theorem, but I was wondering: is there an analogous statement for n-dimensional systems? For instance, is the following true?

Given some restrictions on the functions $f$ and $g \in \Re^n$, let $x_s(t)$ and $y_{s}(t) \in \Re^n$ be solutions to the initial value problems:


$$\dot{x}(t) = f(x,t), x(t_0) = x_0\\ \dot{y}_{1}(t) = g(x,t), y(t_0) = y_{0}$$

Then if $f(x,t)$ is in the convex hull of $g(x,t)$ in some domain $D$ containing $x_0$ and $y_0$ and $x_0$ is in the convex hull of $y_0$, $x_s(t)$ is in the convex hull of $y_s(t) \forall t > t_0$ when the trajectories stay in this domain $D$.

This result seems like it should be true, intuitively, but I'm having trouble finding a formal statement of it.

 

[Only a Mirage]

Sorry -- I was a little hasty in writing this and meant to say the functions $f$ and $g$ map $\Re^2$ to $\Re$, or, in the second possible theorem, map $\Re^{n+1}$ to $\Re^n$.