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

• Published in: Symmetry (Basel) 2023-05, Vol.15 (5), p.1098
• Main Authors: Sun, Linqi, Xu, Hongwen, Ma, Yan
• Format: Article
• Language: English
• Subjects: Algorithms; Approximation; common fixed points; Hilbert space; implicit iterative algorithm; Inequalities; Iterative algorithms; Iterative methods; Methods; nonexpansive mapping; Optimization; Ordinary differential equations; strong convergence; variational inequality; Viscosity; viscosity approximation method
• ISSN: 2073-8994; 2073-8994
• DOI: 10.3390/sym15051098

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.