zbMATH — the first resource for mathematics

Fixed points of nonexpanding maps. (English) Zbl 0177.19101
Let \(f\) be a non-expanding \((\Vert f(x)-f(y)\Vert \le \Vert x-y\Vert)\) mapping of the unit ball in a Hilbert space into itself. Then \(g_k=kf\) \((\vert k\vert<1)\) is a contraction and so has a unique fixed point \(y_k\). The author shows (Theorem 1) that \(\displaystyle\lim_{k\to1,\ \vert k\vert<1} y_k\) exists and is a fixed point of \(f\), moreover it is that unique fixed point \(y\) of \(f\) with minimal norm. The main part of the paper is concerned with iterative procedures for approximating \(y\). Specifically, the author considers sequences \(\{k_n\}\) of real numbers and gives necessary and sufficient conditions (in Theorems 2 and 3, respectively) on such a sequence in order that the sequence of points \(\{z_n\}\) defined recursively by \(z_{n+1}= k_{n+1}f(z_n)\) \((z_0\) being chosen arbitrarily) should converge to \(y\).
Reviewer: A. C. Thompson

47-XX Operator theory
Full Text: DOI
[1] Felix E. Browder, Fixed-point theorems for noncompact mappings in Hilbert space, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 1272 – 1276. · Zbl 0125.35801
[2] Felix E. Browder, Convergence of approximants to fixed points of nonexpansive non-linear mappings in Banach spaces, Arch. Rational Mech. Anal. 24 (1967), 82 – 90. · Zbl 0148.13601 · doi:10.1007/BF00251595 · doi.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.