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
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[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
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