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A general iterative algorithm for nonexpansive mappings in Hilbert spaces. (English) Zbl 1192.47064
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 $$\langle(\gamma f-\mu F)\widetilde x,x-\widetilde x\rangle\leq 0$$, for $$x\in \text{Fix}(T)$$.

##### MSC:
 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47H05 Monotone operators and generalizations 47H06 Nonlinear accretive operators, dissipative operators, etc. 47J20 Variational and other types of inequalities involving nonlinear operators (general)
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