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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 $\left\{{x}_{n}\right\}$ generated by:

$\begin{array}{c}{x}_{0}\in C,\phantom{\rule{4pt}{0ex}}{y}_{n}={\alpha }_{n}{x}_{n}+\left(1-{\alpha }_{n}\right)T{x}_{n},\phantom{\rule{4pt}{0ex}}{\alpha }_{n}\in \left(0,1\right),\\ {C}_{n}=\left\{z\in C:{∥{y}_{n}-z∥}^{2}\le {∥{x}_{n}-z∥}^{2}+\left(1-{\alpha }_{n}\right)\left(k-{\alpha }_{n}\right){∥{x}_{n}-T{x}_{n}∥}^{2}\right\},\\ {Q}_{n}=\left\{z\in C:〈{x}_{n}-z,{x}_{0}-{x}_{n}〉\ge 0\right\},\\ {x}_{n+1}={P}_{{C}_{n}\cap {Q}_{n}}\left({x}_{0}\right),\end{array}$

where $P$ is the metric projection. They show that $\left\{{x}_{n}\right\}$ converges weakly to a fixed point of $T$ (Theorem 3.1), or, respectively, $\left\{{x}_{n}\right\}$ converges strongly to ${P}_{\text{Fix}\left(T\right)}\left({x}_{0}\right)$ (Theorem 4.1).

##### MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47H09 Mappings defined by “shrinking” properties