Forcing Strong Convergence of a Mann-Based Iteration for Nonexpansive and Monotone Operators in a Hilbert Space

Songtao Lv,Sun Young Cho

doi:10.1155/2022/5450521

Abstract

Mann iteration is weakly convergent in infinite dimensional spaces. We, in this paper, use the nearest point projection to force the strong convergence of a Mann-based iteration for nonexpansive and monotone operators. A strong convergence theorem of common elements is obtained in an infinite dimensional Hilbert space. No compact conditions are needed.

Highlights

PreliminariesIn the real world, there are a lot of nonlinear phenomena, which can be modelled into variational inequalities and variational inclusions, such as signal processing, image recovery, and machine learning; see, e.g., [1,2,3,4,5,6,7] and the references therein
Mann iteration is popular for dealing with fixed points of nonexpansive operators
The fixed point solves a monotone variational inequality with S. Another popular regularized method is the hybrid projection method, which was considered by Nakajo and Takahashi [18] for fixed points of nonexpansive mappings first

Summary

Introduction

PreliminariesIn the real world, there are a lot of nonlinear phenomena, which can be modelled into variational inequalities and variational inclusions, such as signal processing, image recovery, and machine learning; see, e.g., [1,2,3,4,5,6,7] and the references therein. Mann iteration is popular for dealing with fixed points of nonexpansive operators. Another popular regularized method is the hybrid projection method, which was considered by Nakajo and Takahashi [18] for fixed points of nonexpansive mappings first.

Results

Conclusion