Hybrid steepest-descent methods with a countable family of nonexpansive mappings for variational inequalities in Hilbert spaces

Abstract

Let H be a Hilbert space and let and be the countable families of nonexpansive mappings of H into itself such that . Assume that F is a nonlinear operator which is κ-Lipschitzian and η-strong monotone on C. In this article, we devise a new iterative scheme {x _n } from an arbitrary initial point x ₀∈H for the countable family of nonexpansive mappings and and prove that {x _n } strongly converges to the unique solution x* of the variational inequality

Applications to constrained generalized pseudoinverse are included.

Keywords:

• nonexpansive mapping
• Banach space
• equilibrium problem
• strong convergence

AMS Subject Classifications:

• 47H05
• 47H09
• 47J25
• 47N10