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:


<br /> \dot{x}(t) = f(x,t), x(t_0) = x_0\\<br /> \dot{y}(t) = g(x,t), y(t_0) = y_0<br />

Then if f(x,t) &gt; g(x,t) in some domain D containing x_0 and y_0 and x_0 \geq y_0, x_s(t) &gt; y_s(t) \forall t &gt; 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:


<br /> \dot{x}(t) = f(x,t), x(t_0) = x_0\\<br /> \dot{y}_{1}(t) = g(x,t), y(t_0) = y_{0}<br />

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 &gt; 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.