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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:

  1. Which Banach space is the natural one to use for functions that map into $\mathbb R^2$?

  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.

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Well, if $E$ is any Banach space, the natural norm on $C(X,E)$ is $\sup_{x \in X} \|f(x)\|_{E}$, where $\|\cdot\|_{E}$ denotes the norm on $E$. Now if $E = \mathbb{R}^2$ I'd probably choose the maximum norm $\left\Vert\begin{pmatrix} x \\y\end{pmatrix}\right\Vert_{\infty} = \max{\{|x|,|y|\}}$, but you can choose your favourite norm on $\mathbb{R}^2$, that is, one which may be better adapted to your problem. What is a bit unclear to me in your question is how your map $G$ is supposed to act on $C(X,\mathbb{R}^2)$. – t.b. May 9 '11 at 18:55
@Theo I edited the question clarifying the nature of $G$. In my particular problem it is just an ugly combination of $F_1$ and $F_2$ evaluated at different points. The norm you suggest in my case should be $\left\Vert \left(\begin{array}{c} F_{1}\left(\cdot\right)\\ F_{2}\left(\cdot\right) \end{array}\right)\right\Vert =\max\left\{ \sup_{y\in\mathbb{R}}\left\{ F_{1}\left(y\right)\right\} ,\sup_{y\in\mathbb{R}}\left\{ F_{2}\left(y\right)\right\} \right\}$. Is that right? – hulp10 May 9 '11 at 19:08
No, it's the other way around. A priori, $X$ is an arbitrary (topological or metric space) and the norm I suggest to use is $\left\Vert\begin{pmatrix} F_1(\cdot) \\F_2(\cdot)\end{pmatrix}\right\Vert = \sup_{x \in X} \max{\{|F_1(x)|,|F_2(x)|\}}$. What is $X$ in your question? Is it $\mathbb{R}$ or $\mathbb{R}^2$? I guess it's the former, but I'm not 100% sure. – t.b. May 9 '11 at 19:14
@Theo In my case is $\mathbb R$, right. As you argue, using a more general space wouldn't change things. I'll try with your suggestion; I am not very familiar with vector-valued norms. – hulp10 May 9 '11 at 19:29
I guess $G_1$ and $G_2$ are bounded functions: $\mathbb{R}^2 \to \mathbb{R}$. This gives a function $G=(G_1,G_2)$. What confuses me a bit is your fixed point equation and your comment afterwards. Writing $F = (F_1, F_2)$, your fixed point equation should become $G \circ F = F$, and your looking for an $F$ satisfying this. Now to apply the fixed point theorem, you should find a closed subset $\mathcal{F}$ of $C(X,\mathbb{R}^2)$ that is preserved by $G$ and $G$ acts contractively. Then you can start with any $H$ in $\mathcal{F}$. Iterating $H \mapsto G \circ H$ gives you the unique fixed point. – t.b. May 9 '11 at 19:36

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