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Zbl 1035.47048
Nakajo, Kazuhide; Takahashi, Wataru
Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups.
(English)
[J] J. Math. Anal. Appl. 279, No. 2, 372-379 (2003). ISSN 0022-247X

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: $$\cases x_0=x\in C,\\ y_n=\alpha_nx_n+ (1-\alpha_n)Tx_n,\ \alpha_n \in [0,a),\ a\in[0,1),\\ C_n=\bigl\{z\in C:\Vert y_n-z\Vert\le\Vert x_n-z \Vert \bigr\},\\ Q_n=\bigl\{z\in C:(x_n-z, x_0-x_n)\ge 0\bigr\},\\ x_{n+1}= P_{C_n\cap Q_n}(x_0),\endcases$$ 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.
[Jarosław Górnicki (Rzeszów)]
MSC 2000:
*47H20 Semigroups of nonlinear operators
47H09 Mappings defined by "shrinking" properties
49M37 Methods of nonlinear programming type

Keywords: real Hilbert space; nonexpansive mapping; metric projection; strong convergence

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