The Fixed Point Problem for Systems of Coordinate-Wise Uniformly Monotone Operators and Applications

We study the fixed point problem for a system of multivariate operators that are coordinate-wise uniformly monotone, 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 the considered problem, together with an approximating iterative scheme, in the setting of partially ordered metric spaces. As an application, we investigate a new abstract multidimensional fixed point problem. To validate our results, we also provide an application to a first-order differential system with periodic boundary value conditions.