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

MSC:

 47-XX Operator theory

Keywords:

functional analysis
Full Text:

References:

 [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
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