A general iterative method of fixed points for equilibrium problems and optimization problems

The purpose of this paper is to present a general iterative scheme as below:and to prove that, if {α _n } and {r _n } satisfy appropriate conditions, then iteration sequences {x _n } and {u _n } converge strongly to a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping and the set of solution of a variational inequality, too. Furthermore, by using the above result, we can also obtain an iterative algorithm for solution of an optimization problem , where h(x) is a convex and lower semicontinuous functional defined on a closed convex subset C of a Hilbert space H. The results presented in this paper extend, generalize and improve the results of Combettes and Hirstoaga, Wittmann, S.Takahashi, Giuseppe Marino, Hong-Kun Xu, and some others. This research is supported by the National Natural Science Foundation of China under Grant No. 10771050.