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!