With $A = (1-2r)I_{q} + rB$, $r = \frac{\tau}{\delta^2}$, $\delta = \frac{1}{q+1}$, give a necessary and sufficient condition on $r$ for the sequence $\left(F_{n}\right)_{n \in \mathbb{N}}$ to be bounded regardless of the choices of $q$ and $F_{0}$.
With $A = (1-2r)I_{q} + rB$, $r = \frac{\tau}{\delta^2}$, $\delta = \frac{1}{q+1}$, give a necessary and sufficient condition on $r$ for the sequence $\left(F_{n}\right)_{n \in \mathbb{N}}$ to be bounded regardless of the choices of $q$ and $F_{0}$.