Krasnoselskii–Mann Viscosity Approximation Method for Nonexpansive Mappings

Abstract

We show that the viscosity approximation method coupled with the Krasnoselskii–Mann iteration generates a sequence that strongly converges to a fixed point of a given nonexpansive mapping in the setting of uniformly smooth Banach spaces. Our result shows that the geometric property (i.e., uniform smoothness) of the underlying space plays a role in relaxing the conditions on the choice of regularization parameters and step sizes in iterative methods.

Keywords

• nonexpansive mapping
• Krasnoselskii–Mann
• viscosity approximation method
• strong convergence
• uniformly smooth Banach space