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Zbl 1174.47059
Song, Yisheng; Chai, Xinkuan
(Song, Yi-sheng; Chai, Xin-kuan)
Strong convergence theorems of viscosity approximation methods for generalized Lipschitz pseudocontractive mappings. (Chinese. English summary)
[J] Acta Math. Sin., Chin. Ser. 51, No. 3, 501-508 (2008). ISSN 0583-1431

Summary: Let $K$ be a nonempty closed convex subset of a Banach space $E$ and $T : K \to K$ be a generalized Lipschitz pseudocontractive mapping. For any fixed Lipschitz strong pseudocontractive mapping $f : K \to K$, let the sequence $\{x_n\}$ be given by $x_1 \in K$, $x_{n+1}=(1-\alpha_n-\beta_n)x_n+\alpha_nf(x_n)+\beta_nTx_n$. It is shown, under appropriate conditions on the sequences of real numbers $\{\alpha_n\}$ and $\{\beta_n\}$, that $\{x_n\}$ strongly converges to some fixed point $x^*$ of $T$ whenever $\{z\in K: \mu_n \|x_n - z\|^2=\inf_{y \in K}\mu_n\|x_n-y\|^2\}\cap F(T)\neq \emptyset$.

MSC 2000:: *47J25 Methods for solving nonlinear operator equations (general)
47H06 Accretive operators, etc. (nonlinear)
47J05 Equations involving nonlinear operators (general)

Keywords: generalized Lipschitz pseudocontractions; viscosity approximations; Banach limits; strong convergence

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