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

摘要

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 descentmethod. Using this method, we can find the common element of the set of fixed points of a nonexpansive mapping, theset of solutions of generalized mixed equilibrium problems, and the set of solutions of variational inequality problems fora relaxed cocoercive mapping in a real Hilbert space. Then, we prove the strong convergence of the proposed iterativescheme to the unique solution of variational inequality. The results presented in this paper generalize and extend somewell-known strong convergence theorems in the literature.