Single-Valued Demicontractive Mappings: Half a Century of Developments and Future Prospects

Abstract

Demicontractive operators form an important class of nonexpansive type mappings whose study led researchers to the creation of some beautiful results in the framework of metric fixed-point theory. This article aims to provide an overview of the most relevant results on the approximation of fixed points of single-valued demicontractive mappings in Hilbert spaces. Subsequently, we exhibit the role of additional properties of demicontractive operators, as well as the main features of the employed iterative algorithms to ensure weak convergence or strong convergence. We also include commentaries on the use of demicontractive mappings to solve some important nonlinear problems with the aim of providing a comprehensive starting point to readers who are attempting to apply demicontractive mappings to concrete applications. We conclude with some brief statements on our view on relevant and promising directions of research on demicontractive mappings in nonlinear settings (metric spaces) and some application challenges.

Keywords

• Hilbert space
• quasi-nonexpansive mapping
• demicontractive mapping
• fixed point
• iterative method
• variational inequality