×

An iterative method for finding common solutions of equilibrium and fixed point problems. (English) Zbl 1141.47040

By combining the iterative scheme for a finite family of nonexpansive mappings due to S. Atsushiba and W. Takahashi [Indian J. Math. 41, No. 3, 435–453 (1999; Zbl 1055.47514)] and the iterative scheme for finding the solution of an equilibrium problem due to S. Plubtieng and R. Punpaeng [J. Math. Anal. Appl. 336, No. 1, 455–469 (2007; Zbl 1127.47053)], the authors present an iterative scheme for finding a common element of the set of solutions of an equilibrium problem and of the set of fixed points of a finite family of nonexpansive mappings in Hilbert space. They prove the strong convergence of the sequences generated by the proposed iterative scheme to a unique solution of a variational inequality. Such type of variational inequality provides the optimality condition for a minimization problem.

MSC:

47J25 Iterative procedures involving nonlinear operators
47J20 Variational and other types of inequalities involving nonlinear operators (general)
49J40 Variational inequalities
47H10 Fixed-point theorems
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Atsushiba, S.; Takahashi, W., Strong convergence theorems for a finite family of nonexpansive mappings and applications, B.N. Prasad birth centenary commemoration volume, Indian J. math., 41, 3, 435-453, (1999) · Zbl 1055.47514
[2] Bauschke, H.H., The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space, J. math. anal. appl., 202, 1, 150-159, (1996) · Zbl 0956.47024
[3] Bauschke, H.H.; Borwein, J.M., On projection algorithms for solving convex feasibility problems, SIAM rev., 38, 3, 367-426, (1996) · Zbl 0865.47039
[4] Blum, E.; Oettli, W., From optimization and variational inequalities to equilibrium problems, Math. student, 63, 1-4, 123-145, (1994) · Zbl 0888.49007
[5] Ceng, L.C.; Cubiotti, P.; Yao, J.C., Strong convergence theorems for finitely many nonexpansive mappings and applications, Nonlinear anal., 67, 1464-1473, (2007) · Zbl 1123.47044
[6] Combettes, P.L., The foundations of set theoretic estimation, Proc. IEEE, 81, 2, 182-208, (1993)
[7] Combettes, P.L., Constrained image recovery in a product space, (), 2025-2028
[8] Combettes, Patrick L.; Hirstoaga, Sever A., Equilibrium programming in Hilbert spaces, J. nonlinear convex anal., 6, 1, 117-136, (2005) · Zbl 1109.90079
[9] Deutsch, F.; Hundal, H., The rate of convergence of Dykstra’s cyclic projections algorithm: the polyhedral case, Numer. funct. anal. optim., 15, 5-6, 537-565, (1994) · Zbl 0807.41019
[10] Goebel, K.; Kirk, W.A., Topics in metric fixed point theory, Cambridge stud. adv. math., vol. 28, (1990), Cambridge University Press Cambridge · Zbl 0708.47031
[11] Kikkawa, M.; Takahashi, W., Weak and strong convergence of an implicit iterative process for a countable family of nonexpansive mappings in Banach spaces, Ann. univ. mariae Curie-sklodowska sect. A, 58, 69-78, (2004) · Zbl 1107.47056
[12] P.E. Mainge, A. Moudafi, Coupling viscosity methods with extragradient algorithm for solving equilibrium problems, J. Nonlinear Convex Anal. (2008), in press · Zbl 1189.90120
[13] Marino, G.; Xu, H.K., A general iterative method for nonexpansive mappings in Hilbert spaces, J. math. anal. appl., 318, 1, 43-52, (2006) · Zbl 1095.47038
[14] Moudafi, A., Viscosity approximation methods for fixed-points problems, J. math. anal. appl., 241, 1, 46-55, (2000) · Zbl 0957.47039
[15] Plubtieng, S.; Punpaeng, R., A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, J. math. anal. appl., 336, 1, 455-469, (2007) · Zbl 1127.47053
[16] Suzuki, T., Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. math. anal. appl., 305, 1, 227-239, (2005) · Zbl 1068.47085
[17] Takahashi, W., Weak and strong convergence theorems for families of nonexpansive mappings and their applications, Ann. univ. mariae Curie-sklodowska, 51, 277-292, (1997) · Zbl 1012.47029
[18] Takahashi, W.; Shimoji, K., Convergence theorems for nonexpansive mappings and feasibility problems, Math. comput. modelling, 32, 1463-1471, (2000) · Zbl 0971.47040
[19] Takahashi, W., Nonlinear functional analysis: fixed point theory and its applications, (2000), Yokohama Publishers Yokohama · Zbl 0997.47002
[20] Takahashi, S.; Takahashi, W., Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. math. anal. appl., 331, 1, 506-515, (2007) · Zbl 1122.47056
[21] Xu, H.K., An iterative approach to quadratic optimization, J. optim. theory appl., 116, 3, 659-678, (2003) · Zbl 1043.90063
[22] Xu, H.K., Viscosity approximation methods for nonexpansive mappings, J. math. anal. appl., 298, 279-291, (2004) · Zbl 1061.47060
[23] Yamada, I.; Ogura, N., Hybrid steepest descent method for the variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings, Numer. funct. anal. optim., 25, 7-8, 619-655, (2004) · Zbl 1095.47049
[24] Yao, Y., A general iterative method for a finite family of nonexpansive mappings, Nonlinear anal., 66, 2676-2687, (2007) · Zbl 1129.47058
[25] Youla, D.C., Mathematical theory of image restoration by the method of convex projections, (), 29-77
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.