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Zbl 1116.47053
Marino, Giuseppe; Xu, Hong-Kun
Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces.
(English)
[J] J. Math. Anal. Appl. 329, No. 1, 336-346 (2007). ISSN 0022-247X

A projection Mann type iterative method, introduced in [{\it K.\,Nakajo} and {\it W.\,Takahashi}, J.~Math.\ Anal.\ Appl.\ 279, No.\,2, 372--379 (2003; Zbl 1035.47048)] and used there to approximate fixed points of nonexpansive mappings, is extended to a more general iterative method, appropriate for approximating fixed points of strict pseudocontractions. Let $C$ be a nonempty closed convex subset of a real Hilbert space and $T:C\to C$ be a $k$-strict pseudocontraction. In the present paper, the authors investigate the sequence $\{x_n\}$ generated by: $$\gather x_0 \in C,\ y_n=\alpha_nx_n+ (1-\alpha_n)Tx_n,\ \alpha_n \in (0,1),\\ C_n=\left\{z\in C:\left\Vert y_n-z\right\Vert ^2 \le \left\Vert x_n-z \right\Vert ^2+(1-\alpha_n)(k-\alpha_n)\left\Vert x_n-Tx_n\right\Vert ^2\right\}, \\ Q_n=\left\{z\in C:\left\langle x_n-z, x_0-x_n \right\rangle \ge 0 \right\}, \\ x_{n+1}= P_{C_n\cap Q_n}(x_0),\endgather$$ where $P$ is the metric projection. They show that $\{x_n\}$ converges weakly to a fixed point of $T$ (Theorem 3.1), or, respectively, $\{x_n\}$ converges strongly to $P_{\text{Fix}(T)}(x_0)$ (Theorem 4.1).
[Vasile Berinde (Baia Mare)]
MSC 2000:
*47J25 Methods for solving nonlinear operator equations (general)
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
47H09 Mappings defined by "shrinking" properties

Keywords: Hilbert space; $k$-strict pseudocontraction; fixed point; projection Mann type iterative method; convergence theorem

Citations: Zbl 1035.47048

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