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Zbl 1180.47040
Cianciaruso, Filomena; Marino, Giuseppe; Muglia, Luigi; Yao, Yonghong
(Yao, Yong-hong)
On a two-step algorithm for hierarchical fixed point problems and variational inequalities.
(English)
[J] J. Inequal. Appl. 2009, Article ID 208692, 13 p. (2009). ISSN 1029-242X/e

The paper is concerned with the variational inequality problem of finding $x^* \in \text{Fix}(T)$ with $\langle (I-S)x^*, x-x^* \rangle \geq 0$ for all $x\in \text{Fix}(T)$, where $T,S: C\to C$ are nonexpansive mappings such that Fix$(T)$, the set of fixed points set of $T$, is nonempty, and $C$ is a closed convex subset of a Hilbert space $H$. Let $f: C \to C$ be a contraction. The authors study convergence properties of the iterative process $x_{n+1}=\alpha_n f(x_n)+(1-\alpha_n) T y_n$, $y_n=\beta_n S x_n+(1-\beta_n) x_n$, where $\alpha_n,\beta_n \in [0,1]$.
[Mikhail Yu. Kokurin (Yoshkar-Ola)]
MSC 2000:
*47J20 Inequalities involving nonlinear operators
49J40 Variational methods including variational inequalities
47J25 Methods for solving nonlinear operator equations (general)

Keywords: fixed points; variational inequalities; contractive mappings; iterative methods

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