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Hi

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

Thank you

1

# fixed point of a particular vector valued function

Hi

I have a function F:R^n->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 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 R^n.) Whatever the answer is, could you give me a reference to some theorem that justifies that?

Thank you