A GENERAL ITERATIVE ALGORITHM FOR EQUILIBRIUM PROBLEMS AND VARIATIONAL INEQUALITY PROBLEMS IN A HILBERT SPACE

In this paper, we introduce a general iterative algorithm for finding a common element of the set of solutions of an equilibrium problem, the set of common fixed points of a finite family of nonexpansive mappings and the set of solutions of the variational inequality for a relaxed cocoercive mapping in a Hilbert space. Then, we prove that the iterative sequence converges strongly to a common element of the three sets. The results obtained in this paper extend and improve the several recent results in this area.

A Viscosity Hybrid Steepest Descent Method for Generalized Mixed Equilibrium Problems and Variational Inequalities for Relaxed Cocoercive Mapping in Hilbert Spaces

We present an iterative method for fixed point problems, generalized mixed equilibrium problems, and variational inequality problems. Our method is based on the so-called viscosity hybrid steepest descent method. Using this method, we can find the common element of the set of fixed points of a nonexpansive mapping, the set of solutions of generalized mixed equilibrium problems, and the set of solutions of variational inequality problems for a relaxed cocoercive mapping in a real Hilbert space. Then, we prove the strong convergence of the proposed iterative scheme to the unique solution of variational inequality. The results presented in this paper generalize and extend some well-known strong convergence theorems in the literature.