Viscosity method for hierarchical variational inequalities and variational inclusions on Hadamard manifolds

Abstract

Abstract This article aims to introduce and analyze the viscosity method for hierarchical variational inequalities involving a ϕ-contraction mapping defined over a common solution set of variational inclusion and fixed points of a nonexpansive mapping on Hadamard manifolds. Several consequences of the composed method and its convergence theorem are presented. The convergence results of this article generalize and extend some existing results from Hilbert/Banach spaces and from Hadamard manifolds. We also present an application to a nonsmooth optimization problem. Finally, we clarify the convergence analysis of the proposed method by some computational numerical experiments in Hadamard manifold.

Keywords

• Viscosity method
• Hierarchical variational inquality
• Variational inclusion problem
• Fixed point problem
• Nonexpansive mapping
• ϕ-contraction mapping