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Iterative methods for strict pseudo-contractions in Hilbert spaces. (English) Zbl 1133.47050
This article deals with two iterative algorithms of finding a common fixed points for $N$ strict pseudo-contractions $\{T_i\}_{i=1}^N$ defined on a closed convex subset $C$ of a real Hilbert space $H$ (an operator $T: C \to C$ is a strict pseudo-contraction, if there exists a constant $0 \le k < 1$ such that $\Vert Tx - Ty\Vert ^2 \le \Vert x - y\Vert ^2 + k\Vert (I - T)x - (I - T)y\Vert ^2$). The first algorithm, called parallel, is defined by the formula $$x_{n+1} = \alpha_nx_n + (1 - \alpha_n) \sum_{i=1}^N \lambda_i^{(n)} T_ix_n, \ x_0 \in C,\ \lambda_i^{(n)} > 0, \ \lambda_1^{(n)} + \cdots + \lambda_N^{(n)} = 1;\tag1$$ the second one, called cyclic, by the formula $$x_{n+1} = \alpha_nx_n + (1 - \alpha) T_{[n]}x_n, \quad x_0 \in C, \quad T_{[n]} = T_i, \ i = n(\text{ mod} \, N), \ 1 \le i \le N.\tag2$$ The main results describe (provided that $F = \bigcap_{i=1}^N \text{Fix} (T_i) \ne \emptyset$) conditions on the control sequence $\{\alpha_n\}$ so that the approximations $x_n$ converge weakly to a common fixed point of $\{T_i\}_{i=1}^N$. At the end of the article, some modifications of algorithms (1) and (2) are proposed; it is proved that approximations $x_n$ for these modified algorithms converge strongly to $P_Fx_0$, where $P_F$ is the nearest point projection from $H$ onto $F$.

MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47H09 Mappings defined by “shrinking” properties 65J15 Equations with nonlinear operators (numerical methods)
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