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Zbl 1192.47064
Tian, Ming
A general iterative algorithm for nonexpansive mappings in Hilbert spaces.
(English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73, No. 3, 689-694 (2010). ISSN 0362-546X

Summary: Let $H$ be a real Hilbert space. Suppose that $T$ is a nonexpansive mapping on $H$ with a fixed point, $f$ is a contraction on $H$ with coefficient $0<\alpha <1$, and $F:H\to H$ is a $k$-Lipschitzian and $\eta$-strongly monotone operator with $k>0$, $\eta>0$. Let $0<\mu<2\eta/k^2$, $0<\gamma<\mu\left(\eta-\frac{\mu k^2}{2}\right)/\alpha=\tau/\alpha$. We proved that the sequence $\{x_n\}$ generated by the iterative method $x_{n+1}=\alpha_n\gamma f(x_n)+(I-\mu\alpha_nF)Tx_n$ converges strongly to a fixed point $\widetilde x\in \text{Fix}(T)$, which solves the variational inequality $\langle(\gamma f-\mu F)\widetilde x,x-\widetilde x\rangle\le 0$, for $x\in \text{Fix}(T)$.
MSC 2000:
*47J25 Methods for solving nonlinear operator equations (general)
47H09 Mappings defined by "shrinking" properties
47H05 Monotone operators (with respect to duality)
47H06 Accretive operators, etc. (nonlinear)
47J20 Inequalities involving nonlinear operators

Keywords: nonexpansive mappings; iterative method; variational inequality; fixed point; projection; viscosity approximation

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