The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space (Q1922986)

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. \textit{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 \(n\in {\mathbb N}\), \(T_n=T_{n\mathbf N}\), \(\lambda_n\in (0,1)\), \(\lambda_n\to 1\), \(a, x_0 \in C\). \textit{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 \textit{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.