A COMPOSITE ITERATIVE METHOD FOR GENERALIZED MIXED EQUILIBRIUM PROBLEMS AND VARIATIONAL INEQUALITY PROBLEMS

In this paper, we introduce a new composite iterative scheme for finding the common element of the set of fixed points of a nonexpansive mapping, the set of solution of generalized mixed equilibrium problems and the set of variational inequality problems for an alpha-inverse-strongly monotone mapping with the viscosity approximation method in a real Hilbert space. We show that the sequence converges strongly to a common element of the above three sets under some parameter controlling conditions. Application to optimization problems which is one of the main result in this work is also given. The results in this paper generalize and improve some recent results of Jung [J. S. Jung, Strong convergence of composite iterative methods for equilibrium problems and fixed point problems. Appl. Math. Comput. 213 (2009), 498-505], Jaiboon et al. [C. Jaiboon, P. Kumam and U.W. Humphries, Convergence theorems by the viscosity approximation methods for equilibrium problems and variational inequality problems. J. Comput. Math. Optim. 5 (2009), 29-56], Su et al. [Y. Su, M. Shang and X. Qin, An iterative method of solution for equilibrium and optimization problems. Nonlinear Anal. 69 (2008), 2709-2719], Yao et al. [Y. Yao, Y.C Liou and R. Chen, Convergence theorems for fixed point problems and variational inequality problems. J. Nonlinear Convex Anal. 9 (2008), 239-248] and connected with Jung [J. S. Jung, Strong convergence of composite iterative methods for equilibrium problems and fixed point problems. J. Comput. Anal. Appl. Vol.12, NO.1-A (2010), 124-140].