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Ofoedu, Eric U.; Shehu, Y.

Convergence analysis for finite family of relatively quasi nonexpansive mappings and systems of equilibrium problems. (English) Zbl 1308.47074

Appl. Math. Comput. 217, No. 22, 9142-9150 (2011).
Summary: A new iterative scheme by hybrid method is constructed. Strong convergence of the scheme to a common element of the set of common fixed points of finite family of relatively quasi-nonexpansive mappings and set of common solutions of a system of equilibrium problems in a uniformly convex and uniformly smooth real Banach space is proved using the properties of generalized \(f\)-projection operator. Our results extend important recent results.

Cited in 10 Documents

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.

Keywords:

generalized \(f\)-projection operator; hybrid method; equilibrium problems; relatively quasi nonexpansive mappings; uniformly convex and uniformly smooth real Banach spaces
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References:

[1] Alber, Y. 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, Y. I., Metric and generalized projection operator in Banach spaces: properties and applications, (Theory and Applications of Nonlinear Operators of Accretive and Monotone Type of Lecture Notes in Pure and Applied Mathematics, Vol. 178 (1996), Springer: Springer New York), 15-50 · Zbl 0883.47083
[3] Blum, E.; Oettli, W., From optimization and variational inequalities to equilibrium problems, Math. Student, 63, 123-145 (1994) · Zbl 0888.49007
[4] Butnariu, D.; Reich, S.; Zaslavski, A. J., Asymptotic behaviour of relatively nonexpansive operators in Banach spaces, J. Appl. Anal, 7, 151-174 (2001) · Zbl 1010.47032
[5] Butnariu, D.; Reich, S.; Zaslavski, A. J., Weak convergence of orbits of nonlinear operator in reflexive Banach spaces, Numer. Funct. Anal. Optim., 24, 489-508 (2003) · Zbl 1071.47052
[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] C.E. Chidume, Geometric properties of Banach spaces and nonlinear iterations, Springer Verlag Series: Lecture Notes in Mathematics Vol. 1965, 2009, XVII, p. 326, ISBN 978-1-84882-189-7.; C.E. Chidume, Geometric properties of Banach spaces and nonlinear iterations, Springer Verlag Series: Lecture Notes in Mathematics Vol. 1965, 2009, XVII, p. 326, ISBN 978-1-84882-189-7. · Zbl 1167.47002
[8] (Chinchuluun, A.; Pardalos, P.; Migdalas, A.; Pitsoulis, L., Pareto Optimality, Game Theory and Equilibria (2008), Springer) · Zbl 1143.91004
[9] Cioranescu, I., Geometry of Banach Spaces Duality Mappings and Nonlinear Problems (1990), Kluwer Academic Dordrecht · Zbl 0712.47043
[10] Combettes, P. L.; Hirstoaga, S. A., Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6, 117-136 (2005) · Zbl 1109.90079
[11] Fan, J. H.; Liu, X.; Li, J. L., Iterative schemes for approximating solutions of generalized variational inequalities in Banach spaces, Nonlinear Anal., 70, 3997-4007 (2009) · Zbl 1219.47110
[12] (Floudas, C. A.; Christodoulos, A.; Pardalos, P. M., Encyclopedia of Optimization, Vol. XXXIV (2009), Springer), 4626, 613 illus. · Zbl 1156.90001
[13] (Giannessi, F.; Maugeri, A.; Pardalos, P. M., Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models, Vol. 58 (2002), Springer) · Zbl 0979.00025
[14] 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
[15] Lin, Lai-jiu, System of generalized vector quasi-equilibrium problems with applications to fixed point theorems for a family of nonexpansive multivalued mappings, J. Global Optim., 34, 1, 15-32 (2006) · Zbl 1098.90086
[16] Li, X.; Huang, N.; O’Regan, D., Strong convergence theorems for relatively nonexpansive mappings in Banach spaces with applications, Comput. Math. Appl., 60, 1322-1331 (2010) · Zbl 1201.65091
[17] Matsushita, S.; Takahashi, W., A strong convergence theorem for relatively nonexpansive mappings in Banach spaces, J. Approx. Theory, 134, 257-266 (2005) · Zbl 1071.47063
[18] Nilsrakoo, W.; Saejung, S., Strong convergence to common fixed points of countable relatively quasi-nonexpansive mappings, Fixed Points Theory Appl. (2008), Article ID 312454 · Zbl 1203.47061
[19] (Pardalos, P. M.; Rassias, T. M.; Khan, A. A., Nonlinear Analysis and Variational Problems (2010), Springer) · Zbl 1178.49001
[20] Qin, X.; Cho, Y. J.; Kang, S. M., Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces, J. Comput. Appl. Math, 225, 20-30 (2009) · Zbl 1165.65027
[21] Takahashi, W., Nonlinear Functional Analysis-Fixed Point Theory and Applications (2000), Yokohama Publishers Inc: Yokohama Publishers Inc Yokohama, in Japanese · Zbl 0997.47002
[22] Takahashi, W., Nonlinear Functional Analysis (2000), Yokohama Publishers: Yokohama Publishers Yokohama · Zbl 0997.47002
[23] Takahashi, W.; Zembayashi, K., Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Anal., 70, 45-57 (2009) · Zbl 1170.47049
[24] Takahashi, W.; Zembayashi, K., Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings, Fixed Point Theory Appl., 11 (2008), Article ID 528476 · Zbl 1187.47054
[25] Wattanawitoon, K.; Kumam, P., Strong convergence theorems by a new hybrid projection algorithm for fixed point problems and equilibrium problems of two relatively quasi nonexpansive mappings, Nonlinear Anal. Hybrid Syst., 3, 11-20 (2009) · Zbl 1166.47060
[26] Wu, K. Q.; Huang, N. J., The generalized f-projection operator with application, Bull. Aust. Math. Soc., 73, 307-317 (2006) · Zbl 1104.47053
[27] Xu, H. K., Inequalities in Banach spaces with applications, Nonlinear Anal., 16, 2, 1127-1138 (1991) · Zbl 0757.46033
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