Viscosity approximations considering boundary point method for fixed-point and variational inequalities of quasi-nonexpansive mappings

Abstract

Existing viscosity approximation schemes have been extensively investigated to solve equilibrium problems, variational inequalities, and fixed-point problems, and most of which contain that contraction is a self-mapping defined on certain bounded closed convex subset C of Hilbert spaces H for standard viscosity approximation. In particular, if the zero point does not belong to C, the standard viscosity approximation cannot be applied to solve the minimum norm fixed point of some nonlinear operators. In this paper, we introduce three generalized viscosity approximation algorithms with boundary point method for quasi-nonexpansive mappings, which overcome this deficiency above. These three proposed algorithms have simple expressions; moreover, they are easy to implement in the actual computation process. Especially, we can find the minimum norm fixed point of quasi-nonexpansive mappings under contraction is a zero operator.

On Convergence Theorems for Generalized Alpha Nonexpansive Mappings in Banach Spaces

“…As an application, fixed point theory of nonexpansive mapping and its generalization has many applications in different fields such as applications of nonexpansive mapping to solve an integral equation (see [18]) and to solve a variational inequality problem (see [19]). Also, there are applications of some classes of generalized nonexpansive mappings like quasi-nonexpansive mappings under contraction to find the minimum norm fixed point and generalized α-nonexpansive mappings to solve split feasibility problem (see [20,21]).…”