Symmetry, Vol. 15, Pages 1098: A New Viscosity Implicit Approximation Method for Solving Variational Inequalities over the Common Fixed Points of Nonexpansive Mappings in Symmetric Hilbert Space

Symmetry, Vol. 15, Pages 1098: A New Viscosity Implicit Approximation Method for Solving Variational Inequalities over the Common Fixed Points of Nonexpansive Mappings in Symmetric Hilbert Space

Symmetry doi: 10.3390/sym15051098

Authors: Linqi Sun Hongwen Xu Yan Ma

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.