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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!

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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.

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