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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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References:

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