Metric projection and convergence theorems for nonexpansive mappings in Hadamard spaces

TL;DR

This paper characterizes metric projections in Hadamard spaces and proves strong convergence of iterative algorithms for nonexpansive mappings, advancing understanding of fixed point theory in non-linear geometric contexts.

Contribution

It provides a new characterization of metric projections in Hadamard spaces and establishes convergence results for iterative algorithms involving nonexpansive mappings.

Findings

Characterization of metric projection via inner product inequality.

Strong convergence of iterative algorithms with perturbations.

Application to fixed point problems in Hadamard spaces.

Abstract

For a nonempty convex subset $C$ of a Hadamard space $X$, it is proved that $u=P_Cx$ if and only if $\langle\overrightarrow{xu},\overrightarrow{uy}\rangle \geqslant0$ for all $y\in C$. As an application of this characterization, we prove strong convergence of two iterative algorithms with perturbations for nonexpansive mappings.