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A general composite explicit iterative scheme of fixed point solutions of variational inequalities for nonexpansive semigroups. (English) Zbl 1218.47105

Summary: We introduce a composite explicit viscosity iteration method for fixed point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces. We prove strong convergence theorems of the composite iterative schemes which solve some variational inequalities under appropriate conditions. Our results extend and improve those announced by S.-H. Li, L.-H. Li and Y.-F. Su [Nonlinear Anal., Theory Methods Appl. 70, No. 9, A, 3065–3071 (2009; Zbl 1177.47075)], S. Plubtieng and R. Punpaeng [Math. Comput. Modelling 48, No. 1–2, 279–286 (2008; Zbl 1145.47308)], S. Plubtieng and R. Wangkeeree [Bull. Korean Math. Soc. 45, No. 4, 717–728 (2008; Zbl 1169.47055)], and many others.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
65K15 Numerical methods for variational inequalities and related problems
47H20 Semigroups of nonlinear operators
47J20 Variational and other types of inequalities involving nonlinear operators (general)
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References:

[1] Browder, F.R., Convergence of approximates to fixed points of nonexpansive nonlinear mappings in Banach spaces, Arch. ration. mech. anal., 24, 82-90, (1967) · Zbl 0148.13601
[2] Reich, S., Strong convergence theorems for resolvent of accretive operators in Banach spaces, J. math. anal. appl., 75, 287-292, (1980) · Zbl 0437.47047
[3] Takahashi, W.; Ueda, Y., On reich’s strong convergence theorems for resolvents of accretive operators, J. math. anal. appl., 104, 546-553, (1984) · Zbl 0599.47084
[4] Baillon, J.B., Un théoréme de type ergodique pour LES contractions non linéaires dans un espace de Hilbert, C.R. acad. sci. Paris Sér., 280, 1511-1514, (1975) · Zbl 0307.47006
[5] Baillon, J.B.; Brézis, H., Une remarque sur le comportement asymptotique des semi-groupes non linéaires, Houston J. math., 2, 5-7, (1976) · Zbl 0318.47039
[6] Plubtieng, S.; Punpaeng, R., Fixed-point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces, Math. comput. modelling, 48, 279-286, (2008) · Zbl 1145.47308
[7] Plubtieng, S.; Wangkeeree, R., A general viscosity approximation method of fixed point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces, Bull. Korean math. soc., 45, 4, 717-728, (2008) · Zbl 1169.47055
[8] Li, S.; Li, L.; Su, Y., General iterative methods for a one-parameter nonexpansive semigroup in Hilbert spaces, Nonlinear anal., 70, 3065-3071, (2009) · Zbl 1177.47075
[9] Li, S.; Li, L.; Su, Y., Composite implicit general iterative process for a nonexpansive semigroup in Hilbert space, Fixed point theory appl., 2008, (2008), Article ID 484050, 13 pages · Zbl 1177.47074
[10] 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
[11] Wangkeeree, R., A general approximation method for nonexpansive semigroups in Hilbert spaces, Far east J. appl. math., 34, 221-232, (2009) · Zbl 1166.47059
[12] Opial, Z., Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. amer. math. soc., 73, 595-597, (1967) · Zbl 0179.19902
[13] Cho, Y.J.; Zhou, H.; Guo, G., Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings, Comput. math. appl., 47, 707-717, (2004) · Zbl 1081.47063
[14] Shimizu, T.; Takahashi, W., Strong convergence to common fixed points of families of nonexpansive mappings, J. math. anal. appl., 211, 71-83, (1997) · Zbl 0883.47075
[15] Xu, H.K., Viscosity approximation method for nonexpansive mappings, J. math. anal. appl., 298, 279-291, (2004) · Zbl 1061.47060
[16] Reich, S., Almost convergence and nonlinear ergodic theorems, J. approx. theory, 24, 269-272, (1978) · Zbl 0404.47032
[17] Reich, S., A note on the Mean ergodic theorem for nonlinear semigroups, J. math. anal. appl., 91, 547-551, (1983) · Zbl 0521.47034
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