Matrix Norm Inequalities

Question

Given $$\Vert A x_0 - {A'} {x_0'} \Vert_2 \leq \beta_1 $$ and $$ \Vert x_0 - x'_0 \Vert_2 \leq \beta_2 $$ where $A \in \mathbb{R}^{n \times n}$, $x_0 \in \mathbb{R}^n $, can we find $\gamma_{min}, \gamma_{max} > 0 $ such that the following relationship holds $$\gamma_{min} \leq \Vert A - A' \Vert_2 \leq \gamma_{max} $$