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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

Cited in: Zbl 1250.47067 Zbl 1237.47075 Zbl 1217.47128 Zbl 1203.47046 Zbl 1197.47074 Zbl 1212.47071 Zbl 1222.47120 Zbl 1220.47114 Zbl 1220.47122 Zbl 1207.47072 Zbl 1203.47068 Zbl 1172.47053 Zbl 1168.47056 Zbl 1134.47052 Zbl 1155.47054 Zbl 1155.47317 Zbl 1124.47046 Zbl 1116.47053 Zbl 1090.47059

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