Banach Contractive Mapping Theorem on Partially Ordered Sets Equipped with Vector Metric

In this paper, left (right) vector metric convergence and left (right) vector metric continuity are defined using the positive and negative parts of a vector-valued metric. These definitions and the obtained new results are applied to the Banach Contractive Mapping Theorem and examined within the framework of order structures, such as partially ordered sets, Archimedean spaces, and Dedekind complete spaces.