I am familiar with the Banach Fixed Point Theorem and I have used it to prove existence and uniqueness of functional equations in Banach spaces like $C(X)$, the space of bounded continuous function $f:X\rightarrow \mathbb R$ with the sup norm $||f||=\sup_{x\in X}|f(x)|$, $X\subset \mathbb R^n$.
I am trying now to show existence and, particularly, uniqueness of the solution to a system of functional equations $F: \mathbb R^2 \rightarrow \mathbb R^2$ that looks like this:
$\left(\begin{array}{c} F_{1}\left(x_{1}\right)\\ F_{2}\left(x_{2}\right) \end{array}\right)=\left(\begin{array}{c} G_{1}\left[F_{1}\left(\cdot\right),F_{2}\left(\cdot\right)\right]\\ G_{2}\left[F_{1}\left(\cdot\right),F_{2}\left(\cdot\right)\right] \end{array}\right)$
Where $G_1: \mathbb R^2 \rightarrow \mathbb R$ and $G_2: \mathbb R^2 \rightarrow \mathbb R$.
In general, $F_{1}$ and $F_{2}$ in the right hand side are not evaluated only at $x_1$ and $x_2$, so the problem cannot be solved pointwise. All the functions behave nicely and I have solved my problem numerically and it converges smoothly (like a contraction).
I have two questions:
Which Banach space is the natural one to use for functions that map into $\mathbb R^2$?
Are there easy to check sufficient conditions for my two dimensional system to be a contraction, like Blackwell's Sufficient Conditions?
I have looked into some fixed-point theory books, but I haven't found anything that directly applies to my problem.