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The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space. (English) Zbl 0956.47024
Let $\left(H,\parallel ·\parallel \right)$ be a real Hilbert space. Suppose ${T}_{1},\cdots ,{T}_{N}$ are non-expansive self-mappings of some closed convex subset $C$ of $H$. (A mapping $T$ is nonexpansive if $\parallel Tx-Ty\parallel ⩽\parallel x-y\parallel$ for all $x,y\in C$.) One possible way to find a common fixed point for the mappings ${T}_{1},\cdots ,{T}_{N}$ 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 $N=1$: ${x}_{n+1}={\lambda }_{n+1}a+\left(1-{\lambda }_{n+1}\right){T}_{n+1}{x}_{n}$ for $n\in ℕ$, ${T}_{n}={T}_{n𝐍}$, ${\lambda }_{n}\in \left(0,1\right)$, ${\lambda }_{n}\to 1$, $a,{x}_{0}\in C$. 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 for nonlinear operators on topological linear spaces 47H09 Mappings defined by “shrinking” properties