On approximation of joint fixed points

TL;DR

This paper explores conditions under which iterative sequences can approximate joint fixed points of a family of mappings on a poset, extending classical fixed point theorems with constructive approximation methods.

Contribution

It provides new conditions on posets and families of mappings that enable approximation of joint fixed points through iterative sequences, extending classical theorems constructively.

Findings

Iterative sequences can approximate joint fixed points under certain conditions.

The paper generalizes classical fixed point theorems to a constructive setting.

Conditions on the poset and mappings ensure convergence of iterative approximations.

Abstract

For a given partially ordered set (poset) and a given family of mappings of the poset into itself, we study the problem of the description of joint fixed points of this family. Well-known Tarski's theorem gives the structure of the set of joint fixed points of isotone automorphisms on a complete lattice. This theorem has several generalizations (see., e.g., Markowsky, Ronse) that weaken demands on the order structure and upgrade in an appropriate manner the assertion on the structure of the set of joint fixed points. However, there is a lack of the statements similar to Kantorovich or Kleene theorems, describing the set of joint fixed points in terms of convergent sequences of the operator degrees. The paper provids conditions on the poset and on the family; these conditions ensure that the iterative sequences of elements of this family approximate the set of joint fixed points. The result obtained develops in a constructive direction the mentioned theorems on joint fixed points.