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 \(x\in \text{Fix}(T)\).


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)
Full Text: DOI


[1] Moudafi, A., Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241, 46-55 (2000) · Zbl 0957.47039
[2] Xu, H. K., Viscosity approximation methods for nonexpansive mapping, J. Math. Anal. Appl., 298, 279-291 (2004) · Zbl 1061.47060
[3] Deutch, F.; Yamada, I., Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings, Numer. Funct. Anal. Optim., 19, 33-56 (1998) · Zbl 0913.47048
[4] Xu, H. K., An iterative approach to quadratic optimiation, J. Optim. Theory Appl., 116, 659-678 (2003) · Zbl 1043.90063
[5] Yamada, I., The hybrid steepest descent for the variational inequality problems over the intersection of fixed points sets of nonexpansive mapping, (Butnariu, D.; Censor, Y.; Reich, S., Inherently Parallel Algorithms in Feasibility and Optimization and Their Application (2001), Elservier: Elservier New York), 473-504 · Zbl 1013.49005
[6] Marino, G.; Xu, H. K., A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 318, 43-52 (2006) · Zbl 1095.47038
[7] Liu, Y., A general iterative method for equilibrium problems and strict pseudo-contrations in Hilbert spaces, Nonlinear Anal. TMA, 71, 4852-4861 (2009) · Zbl 1222.47104
[8] Qin, X.; Shang, M.; Kang, S. M., Strong convergence theorems of modified mann iterative process for strict pseudo-contractions in Hilbert spaces, Nonlinear Anal., 70, 1257-1264 (2009) · Zbl 1225.47107
[9] Xu, H. K., Iterative algorithms for nonlinear operators, J. Lond. Math. Soc., 66, 240-256 (2002) · Zbl 1013.47032
[10] Geobel, K.; Kirk, W. A., (Topics in Metric Fixed Point Theory. Topics in Metric Fixed Point Theory, Cambridge Stud. Adv. Math., vol. 28 (1990), Cambridge Univ. Press), 473-504 · Zbl 0708.47031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.