Iteration and iterative equation on lattices

TL;DR

This paper explores the iteration of maps on lattices and the solutions to polynomial-like iterative equations, using Tarski's fixed point theorem to establish existence results in various Riesz space contexts.

Contribution

It introduces a fixed point approach for iterative equations on lattices without metric structures, extending results to Riesz spaces and their special cases.

Findings

Existence of order-preserving solutions on convex complete sublattices.

Special cases in $ ext{R}^n$ and $ ext{R}$ for semi-continuous and integrable solutions.

Discussion of further Riesz space cases for iterative equations.

Abstract

In this paper we investigate iteration of maps on lattices and the corresponding polynomial-like iterative equation. Since a lattice need not have a metric space structure, neither the Schauder fixed point theorem nor the Banach fixed point theorem is available. Using Tarski's fixed point theorem, we prove the existence of order-preserving solutions on convex complete sublattices of Riesz spaces. Further, in $\mathbb{R}^n$ and $\mathbb{R}$, special cases of Riesz space, we discuss upper semi-continuous solutions and integrable solutions respectively. Finally, we indicate more special cases of Riesz space for discussion on the iterative equation.