A strong convergence theorem on solving common solutions for generalized equilibrium problems and fixed-point problems in Banach space. (English) Zbl 1396.47004

Summary: In this paper, the common solution problem (P1) of generalized equilibrium problems for a system of inverse-strongly monotone mappings \(\{A_k\}^N_{k=1}\) and a system of bifunctions \(\{f_k\}^N_{k=1}\) satisfying certain conditions, and the common fixed-point problem (P2) for a family of uniformly quasi-\(\varphi\)-asymptotically nonexpansive and locally uniformly Lipschitz continuous or uniformly Hölder continuous mappings \(\{S_i\}^\infty_{i=1}\) are proposed. A new iterative sequence is constructed by using the generalized projection and hybrid method, and a strong convergence theorem is proved for approximating a common solution of (P1) and (P2) in Banach space.


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


[1] Zhang F, Su YF: A general iterative method of fixed points for equilibrium problems and optimization problems.J Syst Sci Complex 2009, 22:503-517. · Zbl 1186.47082
[2] Ceng LC, Al-Homidan S, Ansari QH, Yao JC: An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings.J Comput Appl Math 2009, 223:967-974. · Zbl 1167.47307
[3] Zhang SS, Rao RF, Huang JL: Strong convergence theorem for a generalized equilibrium problem and ak-strict pseudocontraction in Hilbert spaces.Appl Math Mech English edition. 2009,30(6):685-694. · Zbl 1177.47086
[4] Peng JW, Yao JC: Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems.Math Comput Model 2009, 49:1816-1828. · Zbl 1171.90542
[5] Peng JW, Yao JC: A viscosity approximation scheme for system of equilibrium problems, nonexpansive mappings and monotone mappings.Nonlinear Anal Theory Methods Appl 2009, 71:6001-6010. · Zbl 1178.47047
[6] Cianciaruso F, Marino G, Muglia L: Iterative methods for equilibrium and fixed point problems for nonexpansive semigroups in Hilbert spaces.J Optim Theory Appl 2010, 146:491-509. · Zbl 1210.47080
[7] Qin XL, Chang SS, Cho YJ: Iterative methods for generalized equilibrium problems and fixed point problems with applications.Nonlinear Anal Real World Appl 2010, 11:2963-2972. · Zbl 1192.58010
[8] Takahashi W, Zembayashi K: Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings.Fixed Point Theory Appl 2008, 2008:11. (Article ID 528476) · Zbl 1187.47054
[9] Takahashi W, Zembayashi K: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces.Nonlinear Anal Theory Methods Appl 2009,70(1):45-57. · Zbl 1170.47049
[10] Chang SS, Lee HWJ, Chan CK: A new hybrid method for solving a generalized equilibrium problem, solving a variational inequality problem and obtaining common fixed points in Banach spaces, with applications.Nonlinear Anal Theory Methods Appl 2010, 73:2260-2270. · Zbl 1236.47068
[11] Chang SS, Kim JK, Wang XR: Modified block iterative algorithm for solving convex feasibility problems in Banach spacesm.J Inequal Appl 2010, 2010:14. (Article ID 869684)
[12] Cioranescu, I.; Hazewinkel, M. (ed.), Geometry of Banach spaces, Duality Mappings and Nonlinear Problems, No. 62 (1990), Dordecht
[13] Alber, YI; Kartosator, AG (ed.), Metric and generalized projection operators in Banach spaces: properities and applications, No. 178, 15-50 (1996), New York
[14] Kamimura S, Takahashi W: Strong convergence of a proxiaml-type algorithm in a Banach space.SIAM J Optim 2002,13(3):938-945. · Zbl 1101.90083
[15] Kohsaka F, Takahashi W: Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces.SIAM J Optim 2008,19(2):824-835. · Zbl 1168.47047
[16] Zhou HY, Gao GL, Tan B: Convergence theorems of a modified hybrid algorithm for a family of quasi-ϕ-asymptotically nonexpansive mappings.J Appl Math Comput 2010, 32:453-464. · Zbl 1203.47091
[17] Matsushita S, Takahashi W: Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces.Fixed Point Theory Appl 2004,2004(1):37-47. · Zbl 1088.47054
[18] Blum E, Oettli W: From optimization and variational inequalities and equilibrium problems.Math Student 1994, 63:123-145. · Zbl 0888.49007
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.