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Wang, Shenghua; Guo, Baohua

New iterative scheme with nonexpansive mappings for equilibrium problems and variational inequality problems in Hilbert spaces. (English) Zbl 1204.65082

J. Comput. Appl. Math. 233, No. 10, 2620-2630 (2010).
The authors introduce a new iterative scheme for finding the common element of the set of solutions of an equilibrium problem and the set of fixed points of an infinite family of nonexpansive mappings in a Hilbert space and prove the strong convergence of the iterative scheme to the unique solution of a variational inequality problem. The authors propose that the iterative scheme of this paper can be studied in Banach spaces also.
In fact the paper will enter into the standard references in the field.
Reviewer: Akrur Behera (Rourkela)

Cited in 12 Documents

MSC:

65K15 Numerical methods for variational inequalities and related problems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47J25 Iterative procedures involving nonlinear operators
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
49J40 Variational inequalities
65J15 Numerical solutions to equations with nonlinear operators

Keywords:

nonexpansive mappings; variational inequality problems; Hilbert spaces; equilibrium problems; iterative scheme; fixed points; strong convergence; Banach spaces
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\textit{S. Wang} and \textit{B. Guo}, J. Comput. Appl. Math. 233, No. 10, 2620--2630 (2010; Zbl 1204.65082)
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References:

[1] Kinderlehrer, D.; Stampacchia, G., An introduction to variational inequalities and their applications, (1980), Academic Press New York, NY · Zbl 0457.35001
[2] Xu, H.K., Viscosity approximation methods for nonexpansive mappings, J. math. anal. appl., 298, 279-291, (2004) · Zbl 1061.47060
[3] Deutsch, 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., Iterative algorithm for nonlinear operators, J. lond. math. soc., 66, 240-256, (2002) · Zbl 1013.47032
[5] Xu, H.K., An iterative approach to quadratic optimization, J. optim. theory appl., 116, 659-678, (2003) · Zbl 1043.90063
[6] Yamada, I., The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings, (), 473-504 · Zbl 1013.49005
[7] Yamada, I.; Ogura, N.; Yamashita, Y.; Sakaniwa, K., Quadratic approximation of fixed points of nonexpansive mappings in Hilbert spaces, Numer. funct. anal. optim., 19, 165-190, (1998) · Zbl 0911.47051
[8] 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
[9] Ceng, L.-C.; Guu, S.M.; Yao, J.C., Hybrid viscosity-like approximation methods for nonexpansive mappings in Hilbert spaces, Comput. math. appl., (2009)
[10] Blum, E.; Oettli, W., From optimization and variational inequalities to equilibrium problems, Math. student, 63, 123-145, (1994) · Zbl 0888.49007
[11] Combettes, P.L.; Hirstoaga, S.A., Equilibrium programming in Hilbert spaces, J. nonlinear convex anal., 6, 117-136, (2005) · Zbl 1109.90079
[12] Takahashi, S.; Takahashi, W., Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. math. anal. appl., 331, 506-515, (2007) · Zbl 1122.47056
[13] Bauschke, H.H., The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space, J. math. anal. appl., 202, 150-159, (1996) · Zbl 0956.47024
[14] Bauschke, H.H.; Borwein, J.M., On projection algorithms for solving convex feasibility problems, SIAM rev., 38, 367-426, (1996) · Zbl 0865.47039
[15] Colao, V.; Marino, G.; Xu, H.-K., An iterative method for finding common solution of equilibrium and fixed problems, J. math. anal. appl., 344, 340-352, (2008) · Zbl 1141.47040
[16] Combettes, P.L., The foundations of set theoretic estimation, Proc. IEEE, 8, 182-208, (1993)
[17] Combettes, P.L., Constrained image recovery in a product space, (), 2025-2028
[18] Deutsch, F.; Hundal, H., The rate of convergence of dykstra’s cyclic projections algorithm: the polyhedral case, Numer. funct. anal. optim., 15, 537-565, (1994) · Zbl 0807.41019
[19] Youla, D.C., Mathematical theory of image restoration by the method of convex projections, (), 29-77
[20] Moudafi, A., Viscosity approximation methods for fixed-points problems, J. math. anal. appl., 241, 46-55, (2000) · Zbl 0957.47039
[21] Goebel, K.; Kirk, W.A., Topics in metric fixed-point theory, (1990), Cambridge University Press Cambridge, England · Zbl 0708.47031
[22] Maingé, P.E., Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces, J. math. anal. appl., 325, 469-479, (2007) · Zbl 1111.47058
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.