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 \(x_ 0\).


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


Zbl 0583.00018