A product of strongly quasi-nonexpansive mappings in Hadamard spaces

TL;DR

This work addresses the convergence of iterative schemes based on products of strongly quasi-nonexpansive mappings in Hadamard (CAT(0)) spaces. It shows that finite products preserve strong quasi-nonexpansiveness and $Δ$-demiclosedness, and that the fixed-point set of a product equals the intersection of the individual fixed-point sets; a $Δ$-convergence theorem for Picard iterations is established, extending to infinite products. The results yield convergent methods for convex function minimization, the minimization of sums of convex functions, and convex feasibility problems in Hadamard spaces, thereby extending proximal-point–style approaches to nonlinear metric geometries. Collectively, these findings provide practical optimization tools in CAT(0) spaces with potential algorithmic implementations for non-Euclidean settings.

Abstract

In this paper, we prove that the product of strongly quasi-nonexpansive $Δ$-demiclosed mappings is also a strongly quasi-nonexpansive orbital $Δ$-demiclosed mapping in Hadamard spaces. Additionally, we establish the $Δ$-convergence theorem for approximating a common fixed point of infinite products of these mappings in Hadamard spaces. Our results have practical applications in convex function minimization, the minimization of the sum of finitely many convex functions, and solving the convex feasibility problem for finitely many sets in Hadamard spaces.

Theorems & Definitions (21)

• Definition 2.1
• Lemma 2.1
• Lemma 2.2
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Proposition 3.1
• proof
• Theorem 3.1