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Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. (English) Zbl 1116.47053
A projection Mann type iterative method, introduced in [K. Nakajo and 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: \[ \begin{gathered} x_0 \in C,\;y_n=\alpha_nx_n+ (1-\alpha_n)Tx_n,\;\alpha_n \in (0,1),\\ C_n=\left\{z\in C:\left\| y_n-z\right\| ^2 \leq \left\| x_n-z \right\| ^2+(1-\alpha_n)(k-\alpha_n)\left\| x_n-Tx_n\right\| ^2\right\}, \\ Q_n=\left\{z\in C:\left\langle x_n-z, x_0-x_n \right\rangle \geq 0 \right\}, \\ x_{n+1}= P_{C_n\cap Q_n}(x_0),\end{gathered} \] 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).

MSC:
47J25 Iterative procedures involving nonlinear operators
47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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