Some new extragradient-like methods for generalized equilibrium problems, fixed point problems and variational inequality problems

Abstract

In this paper, we introduce two iterative schemes by extragradient-like methods for finding a common element of the set of solutions of a generalized equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of the variational inequality for a monotone, Lipschitz-continuous mapping in a Hilbert space. We obtain a strong convergence theorem and a weak convergence theorem for the sequences generated by these processes. Based on these two results, we also get some new and interesting results. The results in this paper generalize and extend some well-known strong convergence theorems and weak convergence theorems in the literature.

Keywords:

• generalized equilibrium problem
• extragradient-like method
• nonexpansive mapping
• monotone mapping
• variational inequality
• strong convergence
• weak convergence
• fixed point

Acknowledgements

The authors would like to express their thanks to the referees for helpful suggestions. This research was supported by the National Natural Science Foundation of China (Grant no. 10771228 and Grant no. 10831009), the Science and Technology Research Project of Chinese Ministry of Education (Grant no. 206123), the Education Committee project Research Foundation of Chongqing (Grant no. KJ070816) and the grant NSC 96-2119-M-110-001.