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On approximating fixed points for nonexpansive maps. (English) Zbl 0793.47052

Summary: Let \(K\) be a closed convex subset of a Hilbert space \(H\) and let \(f: K\to H\) be a nonexpansive map, \(z\) a point in \(K\), \(f_ t(x):= tf(x)+ (1-t)z\) \((t\in (0,1))\) and \(P: H\to K\) the metric projection. We study the behaviour of the fixed points \(x_ t\) and \(y_ t\) of the maps \((Pf)_ t\) and \(Pf_ t\) respectively and we show that if \(f\) has fixed points, then \(\lim_{t\to 1^ -} x_ t= \lim_{t\to 1^ -} y_ t= y\), where \(y\) is the fixed point of \(f\) that is closest to \(z\).

MSC:

47H10 Fixed-point theorems
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