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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 $$0 \leq k < 1$$ such that $$\| Tx - Ty\| ^2 \leq \| x - y\| ^2 + k\| (I - T)x - (I - T)y\| ^2$$). The first algorithm, called parallel, is defined by the formula $x_{n+1} = \alpha_nx_n + (1 - \alpha_n) \sum_{i=1}^N \lambda_i^{(n)} T_ix_n, \;x_0 \in C,\;\lambda_i^{(n)} > 0, \;\lambda_1^{(n)} + \cdots + \lambda_N^{(n)} = 1;\tag{1}$ the second one, called cyclic, by the formula
$x_{n+1} = \alpha_nx_n + (1 - \alpha) T_{[n]}x_n, \quad x_0 \in C, \quad T_{[n]} = T_i, \;i = n(\text{ mod} \, N), \;1 \leq i \leq N.\tag{2}$ The main results describe (provided that $$F = \bigcap_{i=1}^N \text{Fix} (T_i) \neq \emptyset$$) 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 involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 65J15 Numerical solutions to equations with nonlinear operators
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