The fixed point problem for systems of coordinate-wise monotone operators and applications

TL;DR

This paper investigates fixed point problems for systems of coordinate-wise monotone operators within quasi-ordered sets, providing a unified approach and criteria for existence and uniqueness, with applications to differential systems.

Contribution

It introduces a unified framework linking coordinate-wise monotone operators to mixed monotone operators, simplifying the analysis of fixed points in complex systems.

Findings

Established equivalence between coordinate-wise monotone systems and mixed monotone operators.

Provided existence and uniqueness criteria in partially ordered metric spaces.

Applied the theory to differential systems with periodic boundary conditions.

Abstract

We study the fixed point problem for a system of multivariate operators that are coordinate-wise monotone (i.e., nondecreasing or nonincreasing in each of the variables, independently), in the setting of quasi-ordered sets. We show that this problem is equivalent to the fixed point problem for a mixed monotone operator that can be explicitly constructed. As a consequence, we obtain a criterion for the existence and uniqueness of solution to our problem, in the setting of partially ordered metric spaces. To validate our results, we provide an application to a first-order differential system with periodic boundary value conditions. A direct consequence that follows from our paper is that all the separate recent developments on the subject of tripled, quadrupled or multidimensional fixed points are but particular aspects of a single, unified and much simpler approach.