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



fixed point of a particular vector valued function
Your question is stated as a Conjecture just before Theorem 2.1.5 in the book
MR1015711 
Belitskiĭ, G. R. and Lyubich, Yu. I.
Matrix norms and their applications.
Birkhäuser Verlag, Basel, 1988.


