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

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.