## On approximating fixed points.(English)Zbl 0597.47035

Nonlinear functional analysis and its applications, Proc. Summer Res. Inst., Berkeley/Calif. 1983, Proc. Symp. Pure Math. 45/2, 393-395 (1986).
[For the entire collection see Zbl 0583.00018.]
The authors give the following result.
Theorem. Let H be a real Hilbert space and C a closed, convex subset of H. Let f:C$$\to H$$ be a nonexpansive mapping with f(C) bounded and f($$\partial C)\subset C$$. Suppose $$0\in C$$. Let $$f_ k(x)=kf(x)+(1- k)x_ 0$$ for some $$x_ 0\in C$$ and $$0<k<1$$, $$k\to 1$$, and let $$f_ kx_ k=x_ k$$. Then $$x_ n$$ converges strongly to $$y_ 0$$, where $$y_ 0$$ is the fixed point of f closest to $$x_ 0$$.

### MSC:

 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 54H25 Fixed-point and coincidence theorems (topological aspects)

### Keywords:

nonexpansive mapping; fixed point

Zbl 0583.00018