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Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. (English) Zbl 1035.47048
Let \(C\) be a nonempty closed convex subset of a real Hilbert space and \(T:C\to C\) be a nonexpansive mapping. In the present paper, the authors investigate the sequence \(\{x_n\}\) generated by: \[ \begin{cases} x_0=x\in C,\\ y_n=\alpha_nx_n+ (1-\alpha_n)Tx_n,\;\alpha_n \in [0,a),\;a\in[0,1),\\ C_n=\bigl\{z\in C:\| y_n-z\|\leq\| x_n-z \| \bigr\},\\ Q_n=\bigl\{z\in C:(x_n-z, x_0-x_n)\geq 0\bigr\},\\ x_{n+1}= P_{C_n\cap Q_n}(x_0),\end{cases} \] where \(P\) is the metric projection. They show that \(\{x_n\}\) converges strongly to \(P_{\text{Fix}(T)}(x_0)\) by the hybrid method which is used in mathematical programming and obtain a strong convergence theorem for a family of nonexpansive mappings in a real Hilbert space.

MSC:
47H20 Semigroups of nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
49M37 Numerical methods based on nonlinear programming
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