Hi

I have a function $F:\mathbb{R} ^ n\rightarrow \mathbb{R}^n$ for which I know there exist a unique fixed point $x ^ *$ (say). I also know that the Jacobian of $F$ at each point $x$ in $\mathbb{R} ^ n$ has all of its eigenvalues in $[0,1)$ (but they are different for each $x$). Are these facts enough for me to say that the iterative sequence $x _ {n+1} = F(x_ n)$ converges to $x ^ *$ independently of the initial point $x_ 0$? (I know that if $x_0$ is close enough to $x ^ *$ then the sequence coverges but my question concerns any $x_0$ in $\mathbb{R} ^ n$.) Whatever the answer is, could you give me a reference to some theorem that justifies that?

Thank you

realand in $[0,1)$, or just their modulus is in $[0,1)$? $\endgroup$ – Pietro Majer Nov 28 '12 at 17:10