A New Viscosity Implicit Approximation Method for Solving Variational Inequalities over the Common Fixed Points of Nonexpansive Mappings in Symmetric Hilbert Space

In this paper, based on the viscosity approximation method and the hybrid steepest-descent iterative method, a new implicit iterative algorithm is presented for finding the common fixed points set of a finite family of nonexpansive mappings in a reflexive Hilbert space, which is called a symmetric space. We prove that the sequence generated by this new implicit rule strongly converges to the unique solution of a class of variational inequalities under certain appropriate conditions of the parameters. Moreover, we also study the applications to a broader family of strictly pseudo-contractive mappings and generalized equilibrium problems that involve several variational inequality problems, optimization problems, and fixed-point problems. Finally, numerical results are provided to clarify the stability and effectiveness of the algorithm and to compare with some existing iterative algorithms.