×

Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. (English) Zbl 1165.65027

The purpose of the paper is to introduce hybrid projection algorithms to find a common element of the set of common fixed points of two quasi-\(\phi\)-nonexpensive mappings and the set of solutions of an equilibrium problem in the framework of Banach spaces.

MSC:

65J15 Numerical solutions to equations with nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
47J25 Iterative procedures involving nonlinear operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alber, Ya. I.; Reich, S., An iterative method for solving a class of nonlinear operator equations in Banach spaces, Panamer. Math. J., 4, 39-54 (1994) · Zbl 0851.47043
[2] Alber, Ya. I., Metric and generalized projection operators in Banach spaces: Properties and applications, (Kartsatos, A. G., Theory and Applications of Nonlinear Operators of Accretive and Monotone Type (1996), Marcel Dekker: Marcel Dekker New York), 15-50 · Zbl 0883.47083
[3] Butnariu, D.; Reich, S.; Zaslavski, A. J., Asymptotic behavior of relatively nonexpansive operators in Banach spaces, J. Appl. Anal., 7, 151-174 (2001) · Zbl 1010.47032
[4] Butnariu, D.; Reich, S.; Zaslavski, A. J., Weak convergence of orbits of nonlinear operators in reflexive Banach spaces, Numer. Funct. Anal. Optim., 24, 489-508 (2003) · Zbl 1071.47052
[5] Blum, E.; Oettli, W., From optimization and variational inequalities to equilibrium problems, Math. Student, 63, 123-145 (1994) · Zbl 0888.49007
[6] Censor, Y.; Reich, S., Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization, Optimization, 37, 323-339 (1996) · Zbl 0883.47063
[7] Cioranescu, I., Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems (1990), Kluwer: Kluwer Dordrecht · Zbl 0712.47043
[8] Ceng, L. C.; Yao, J. C., Hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings, Appl. Math. Comput., 198, 729-741 (2008) · Zbl 1151.65058
[9] Ceng, L. C.; Yao, J. C., A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. Comput. Appl. Math., 214, 186-201 (2008) · Zbl 1143.65049
[10] Combettes, P. L.; Hirstoaga, S. A., Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6, 117-136 (2005) · Zbl 1109.90079
[11] 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
[12] Kamimura, S.; Takahashi, W., Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. Optim., 13, 938-945 (2002) · Zbl 1101.90083
[13] Matsushita, S. Y.; Takahashi, W., A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. Approx. Theory, 134, 257-266 (2005) · Zbl 1071.47063
[14] Matsushita, S.; Takahashi, W., Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces, Fixed Point Theory Appl., 2004, 37-47 (2004) · Zbl 1088.47054
[15] Moudafi, A., Second-order differential proximal methods for equilibrium problems, J. Inequal. Pure Appl. Math., 4, art. 18 (2003) · Zbl 1175.90413
[16] Plubtieng, S.; Punpaeng, R., A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings, Appl. Math. Comput., 197, 548-558 (2008) · Zbl 1154.47053
[17] Plubtieng, S., Rattanaporn Punpaeng, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl., 336, 455-469 (2007) · Zbl 1127.47053
[18] Qin, X.; Shang, M.; Su, Y., A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, Nonlinear Anal. (2007) · Zbl 1158.47317
[19] Qin, X.; Shang, M.; Su, Y., Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems, Math. Comput. Modelling. (2008) · Zbl 1187.65058
[20] Reich, S., A weak convergence theorem for the alternating method with Bregman distance, (Kartsatos, A. G., Theory and Applications of Nonlinear Operators of Accretive and Monotone Type (1996), Marcel Dekker: Marcel Dekker New York), 313-318 · Zbl 0943.47040
[21] Su, Y.; Shang, M.; Qin, X., An iterative method of solution for equilibrium and optimization problems, Nonlinear Anal. (2007)
[22] Takahashi, W., Nonlinear Functional Analysis (2000), Yokohama-Publishers · Zbl 0997.47002
[23] 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
[24] Takahashi, W.; Zembayashi, K., Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Anal. (2008)
[25] Tada, A.; Takahashi, W., Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem, J. Optim. Theory Appl., 133, 359-370 (2007) · Zbl 1147.47052
[26] Yao, Y.; Noor, M. A.; Liou, Y. C., On iterative methods for equilibrium problems, Nonlinear Anal. (2008)
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.