Nakajo, Kazuhide; Takahashi, Wataru Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. (English) Zbl 1035.47048 J. Math. Anal. Appl. 279, No. 2, 372-379 (2003). 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. Reviewer: Jarosław Górnicki (Rzeszów) Cited in 38 ReviewsCited in 387 Documents MSC: 47H20 Semigroups of nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. 49M37 Numerical methods based on nonlinear programming Keywords:real Hilbert space; nonexpansive mapping; metric projection; strong convergence × Cite Format Result Cite Review PDF Full Text: DOI References: [1] S. Atsushiba, W. Takahashi, A weak convergence theorem for nonexpansive semigroups by the Mann iteration process in Banach spaces, in: W. Takahashi, T. Tanaka (Eds.), Nonlinear Analysis and Convex Analysis World Scientific, Singapore, pp. 102-109; S. Atsushiba, W. Takahashi, A weak convergence theorem for nonexpansive semigroups by the Mann iteration process in Banach spaces, in: W. Takahashi, T. Tanaka (Eds.), Nonlinear Analysis and Convex Analysis World Scientific, Singapore, pp. 102-109 · Zbl 1003.47051 [2] Bruck, R. E., On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces, Israel J. Math., 38, 304-314 (1981) · Zbl 0475.47037 [3] Halpern, B., Fixed points of nonexpanding maps, Bull. Amer. Math. Soc., 73, 957-961 (1967) · Zbl 0177.19101 [4] Kamimura, S.; Takahashi, W., Approximating solutions of maximal monotone operators in Hilbert spaces, J. Approx. Theory, 106, 226-240 (2000) · Zbl 0992.47022 [5] Martinet, B., Regularisation d’inequations variationnelles par approximations successives, Rev. Franc. Inform. Rech. Opér., 4, 154-159 (1970) · Zbl 0215.21103 [6] Opial, Z., Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73, 591-597 (1967) · Zbl 0179.19902 [7] Rockafellar, R. T., Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14, 877-898 (1976) · Zbl 0358.90053 [8] Shimizu, T.; Takahashi, W., Strong convergence to common fixed points of families of nonexpansive mappings, J. Math. Anal. Appl., 211, 71-83 (1997) · Zbl 0883.47075 [9] Shioji, N.; Takahashi, W., Strong convergence theorems for continuous semigroups in Banach spaces, Math. Japon., 50, 57-66 (1999) · Zbl 0940.47047 [10] Solodov, M. V.; Svaiter, B. F., Forcing strong convergence of proximal point iterations in a Hilbert space, Math. Programming Ser. A, 87, 189-202 (2000) · Zbl 0971.90062 [11] Takahashi, W., Nonlinear Functional Analysis (2000), Yokohama Publishers: Yokohama Publishers Yokohama · Zbl 0997.47002 [12] Wittmann, R., Approximation of fixed points of nonexpansive mappings, Arch. Math., 58, 486-491 (1992) · Zbl 0797.47036 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.