Let $E$ a Banach space ($E$ is the space of continuous functions on $[0,T]$ for my case). Let $F, G: E\times E\to E$ be contraction maps of contraction constant $\epsilon>0$. Given $b\in\mathbb R$, consider the map

\begin{eqnarray} (F,G_b) : E\times E &\to& E\times E \\ (x,y) &\mapsto& (F(x,y),G(x,y)+bx). \end{eqnarray}

For any $b$, can we always find $\epsilon>0$ small enough s.t.

- $(F,G_b)$ admits a fixed point?
- $(F,G_b)$ admits a unique fixed point?
- If not, do we have some conditions to ensure the existence/uniqueness?

Any answer, comments and references are highly appreciated!

PS : If it helps, we may use the norm $\|x\|=\max_{0\le t\le T}|x(t)|$ for $x\in E$.