Suppose we have a function $F: [a,b]^n \to \mathcal{M}\_{n \times n }(\mathbb{R})$ where $\mathcal{M}\_{n \times n }(\mathbb{R})$ is the space of $n \times n$ real matrices, a compact set $B \subset \mathbb{R}^n$, a constant $C \in \mathbb{R}$. What kind of $F$ satisfies the following condition
$$\Vert F(x\_1) y\_1 - F(x\_2) y\_2 \Vert\_2 \le C \Vert y\_1 - y\_2 \Vert\_2, \forall x\_1, x\_2 \in [a,b]^n, \forall y\_1, y\_2 \in B$$


If $F$ is constantly a contraction the problem becomes trivial, but what happens when $F$ varies? A simpler question I am interested in is, when $A, B \in \mathcal{M\_{n \times n}(\mathbb{R})}$, what conditions must they satisfy such that
$$\Vert A y\_1 - B y\_2 \Vert\_2 \le C \Vert y\_1 - y\_2 \Vert\_2, \forall y\_1, y\_2 \in B$$
Are there any previous work on problems like this? I would really appreciate some book/paper references!



A question on a special "metric"
Taking $y\_1=y\_2=y$, we see that $F(x\_1)y=F(x\_2)y\,$ for all $x\_1,x\_2$ in $[a,b]^n$ and for all $y\in B$, and hence for all $y\in \text{span}\,B$. In particular, if $B$ spans $\mathbb R^n$, then the matrix $F(x)$ does not depend on $x$, and in such a case the problem becomes "tirvial", as you said. 


