ITERATIVE METHODS FOR GENERALIZED EQUILIBRIUM PROBLEMS AND NONEXPANSIVE MAPPINGS Cho, Sun-Young; Kang, Shin-Min; Qin, Xiaolong;
In this paper, a composite iterative process is introduced for a generalized equilibrium problem and a pair of nonexpansive mappings. It is proved that the sequence generated in the purposed composite iterative process converges strongly to a common element of the solution set of a generalized equilibrium problem and of the common xed point of a pair of nonexpansive mappings.
Existence of solutions for generalized equilibrium problem in G-convex space, Computers & Mathematics with Applications, 2011, 62, 9, 3404
E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium
problems, Math. Student 63 (1994), no. 1-4, 123-145.
F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings
in Hilbert space, J. Math. Anal. Appl. 20 (1967), 197-228.
L. C. Ceng and J. C. Yao, Hybrid viscosity approximation schemes for equilibrium
problems and fixed point problems of infinitely many nonexpansive mappings, Appl.
Math. Comput. 198 (2008), no. 2, 729-741.
O. Chadli, N. C. Wong and J. C. Yao, Equilibrium problems with applications to eigen-value problems, J. Optim. Theory Appl. 117 (2003), no. 2, 245-266.
S. S. Chang, H. W. Joseph Lee and C. K. Chan, A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to
optimization, Nonlinear Anal. 70 (2009), no. 9, 3307-3319.
J. Chen, L. Zhang and T. Fan, Viscosity approximation methods for nonexpansive mappings and monotone mappings, J. Math. Anal. Appl. 334 (2007), no. 2, 1450-1461.
V. Colao, G. Marino and H. K. Xu, An iterative method for finding common solutions of
equilibrium and fixed point problems, J. Math. Anal. Appl. 344 (2008), no. 1, 340-352.
P. L. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J.
Nonlinear Convex Anal. 6 (2005), no. 1, 117-136.
H. Iiduka and W. Takahashi, Strong convergence theorems for nonexpansive mappings
and inverse-strongly monotone mappings, Nonlinear Anal. 61 (2005), no. 3, 341-350.
S. M. Kang, S. Y. Cho and Z. Liu, Convergence of iterative sequences for generalized
equilibrium problems involving inverse-strongly monotone mappings, J. Inequal. Appl.
2010 (2010), Article ID 827082, 16 pages.
A. Moudafi and M. Thera, Proximal and dynamical approaches to equilibrium problems, Ill-posed variational problems and regularization techniques (Trier, 1998), 187-201, Lecture Notes in Econom. and Math. Systems, 477, Springer, Berlin, 1999.
Z. Opial, Weak convergence of the sequence of successive approximation for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 561-597.
S. Plubtieng and R. Punpaeng, A new iterative method for equilibrium problems and
fixed point problems of nonexpansive mappings and monotone mappings, Appl. Math.
Comput. 197 (2008), no. 2, 548-558.
X. Qin, Y. J. Cho and S. M. Kang, Convergence theorems of common elements for
equilibrium problems and fixed point problems in Banach spaces, J. Comput. Appl.
Math. 225 (2009), no. 1, 20-30.
X. Qin, M. Shang and Y. Su, Strong convergence of a general iterative algorithm for
equilibrium problems and variational inequality problems, Math. Comput. Modelling 48
(2008), no. 7-8, 1033-1046.
T. Suzuki, Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl.
305 (2005), no.1, 227-239.
S. Takahashi and W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007), no.
S. Takahashi and W. Takahashi, Strong convergence theorem for a generalized equilibrium problem and a non-expansive mapping in a Hilbert space, Nonlinear Anal. 69 (2008), no. 3, 1025-1033.
H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. (2) 66
(2002), no. 1, 240-256.