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Zbl 1121.65064
Yao, Yonghong; Yao, Jen-Chih
On modified iterative method for nonexpansive mappings and monotone mappings.
(English)
[J] Appl. Math. Comput. 186, No. 2, 1551-1558 (2007). ISSN 0096-3003

Let $C$ be a nonempty, closed, convex subset of a real Hilbert space $H$ and let $A:C\to H$ be an inverse-strongly monotone mapping. Moreover, let $S:C\to C$ be nonexpansive. The authors study a new class of iteration schemes for finding a fixed-point $x^*$ of $S$ that also fulfills the variational inequality $\langle Ax^*, v-x^*\rangle \ge 0$ for all $v\in C$. Results on the strong convergence of the sequence of iterates are proven. Also the case $A=I-T$ with a pseudocontractive mapping $T:C\to C$ is considered.
[Etienne Emmrich (Berlin)]
MSC 2000:
*65J15 Equations with nonlinear operators (numerical methods)
47H05 Monotone operators (with respect to duality)
47H09 Mappings defined by "shrinking" properties
47J25 Methods for solving nonlinear operator equations (general)

Keywords: nonexpansive mapping; monotone mapping; fixed point; variational inequality; iteration; convergence; Hilbert space

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