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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. {\it 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 $n\in {\Bbb N}$, $T_n=T_{n\bold N}$, $\lambda_n\in (0,1)$, $\lambda_n\to 1$, $a, x_0 \in C$. {\it 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 {\it 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.

47H10Fixed-point theorems for nonlinear operators on topological linear spaces
47H09Mappings defined by “shrinking” properties
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