Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. (English) Zbl 1035.47048

Nakajo, Kazuhide; Takahashi, Wataru

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)

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

References:

[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
[12]	Wittmann, R., Approximation of fixed points of nonexpansive mappings, Arch. Math., 58, 486-491 (1992) · Zbl 0797.47036