Let $T, T' \in \mathbb{R}^{n \times n}$ and contractions. Assume that $J, J' \in \mathbb{R}^n$ are the respective fixed points $J' = T'J'$ and $J = TJ$.
Further assuming that the respective fixed points are close in some sense, e.g. $$ \lVert J - J' \rVert < \epsilon, $$ I wonder if there is a bound on the respective differences of the iterates initialised at some point $g$: $$ \lVert T^kg - T'^kg \rVert < \delta_k. $$ I am looking for some form of $\delta_k$, involving $\epsilon$, the distance of $g$ from the fixed points, or something along those lines. The case $k=1$ is of special interest to me. The exact form of the norms (weighted sup norm, L1, L2, ...) I don't care too much about.
I basically need to show that if we do a single iteration with the wrong operator $T'$ instead of $T$, we don't care too much as long as their respective fixed points are close. Intuitively, this should make sense to me if the initialisation is far away from the fixed points, because then the iterations should go into the same direction.
From triangular inequality we obtain $\| T^kg - T'^kg \| = \| T^kg-J+J-J'+J'-T'^kg \|\leq \| T^kg-J\|+\|J-J'\|+\|J'-T'^kg \|.$
Since you assume that $\lVert J - J' \rVert < \epsilon$, $T$ contracts by some positive $\delta$ and $T'$ contracts by some positive $\gamma$. Then $$\| T^kg - T'^kg \| \leq \delta^k \| g-J\|+\epsilon+ \gamma^k\|J'-g \|$$ But I doubt that you can make the bound dependent only on $k$, and not on $g$ too, without further assumptions.