User nima - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T06:40:37Z http://mathoverflow.net/feeds/user/12838 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/54846/converse-of-the-banach-fixed-point-theorem Converse of the Banach fixed point theorem Nima 2011-02-09T04:16:00Z 2011-02-23T07:22:16Z <p>Let $f:X \to X$ ($X = R^d$ for some $d$) be a mapping such that $f^n (x) \to x^\ast$ for all $x \in X$ as $n \to \infty$ ($x^\ast$ is unique). Can we say anything about the spectral structure of the gradient matrix $\nabla f (x^\ast)$ ? Do we know that the spectral or the operator norm of this matrix is less than one?</p>