Abstract

In this paper, we introduce and consider a new class of equilibrium problems and variational inequalities, which is called the mixed equilibrium-variational inequality. We use the auxiliary principle technique to suggest and analyze some explicit and proximal-point iterative methods for solving the mixed equilibrium-variational inequalities. Convergence of these iterative methods is proved under very mild and suitable assumptions. Several special cases are also considered. Results proved in this paper can be viewed as refinement and improvement of the previously known and new results. The ideas and techniques of this paper may stimulate further research in this dynamic area of pure and applied sciences.

Key words: Mixed variational inequality, equilibrium problems, convergence, auxiliary principle technique.