## 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

### Keywords:

nonexpansive map; fixed points