The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space.(English)Zbl 0956.47024

Let $$(H,\|\cdot\|)$$ be a real Hilbert space. Suppose $$T_1,\dots,T_N$$ are non-expansive self-mappings of some closed convex subset $$C$$ of $$H$$. (A mapping $$T$$ is nonexpansive if $$\|Tx-Ty\|\leqslant \|x-y\|$$ for all $$x,y\in C$$.) One possible way to find a common fixed point for the mappings $$T_1,\dots, T_N$$ is to construct a sequence which will converge to the desired point. B. Halpern [Bull. Am. Math. Soc. 73, 957-961 (1967; Zbl 0177.19101)] suggested the following algorithm for $$N=1$$: $$x_{n+1}=\lambda _{n+1}a+(1-\lambda _{n+1})T_{n+1} x_n$$ for $$n\in {\mathbb N}$$, $$T_n=T_{n\mathbf N}$$, $$\lambda_n\in (0,1)$$, $$\lambda_n\to 1$$, $$a, x_0 \in C$$. P.-L. Lions [C. R. Acad. Sci., Paris, Sér. A 284, 1357-1359 (1977; Zbl 0349.47046)] investigated the general case. However the restrictions which they imposed on $$\lambda_n$$ are difficult to verify. Recently R. Wittmann [Arch. Math. 58, No. 5, 486-491 (1992); Zbl 0797.47036] extended the class of admissible sequences $$\lambda_n$$ (for Halpern case $$N=1$$). In this paper the author improves results of Wittmann and Lions and established good assumptions on $$\lambda_n$$ under which the sequence is convergent.

MSC:

 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
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