Compositions of Resolvents: Fixed Points Sets and Set of Cycles

TL;DR

This paper explores the fixed point sets and cycles of compositions of resolvent operators in Hilbert spaces, using Attouch Théra duality to formulate these cycles as solutions to fixed point equations.

Contribution

It introduces a novel formulation linking cycles of resolvent compositions to fixed point sets via duality, advancing understanding in operator theory.

Findings

Cycles can be characterized as fixed points of certain operators

The relationship between cycles and fixed point sets is clarified

New fixed point equations for resolvent compositions are established

Abstract

In this paper, we investigate the cycles and fixed point sets of compositions of resolvents using Attouch Th\'era duality. We demonstrate that the cycles defined by the resolvent operators can be formulated in Hilbert space as the solution to a fixed point equation. Furthermore, we introduce the relationship between these cycles and the fixed point sets of the composition of resolvents.