On coincidence points for vector mappings

For mappings acting in the product of metric spaces we propose a concept of vector covering. This concept is a natural extension of the notion of covering for mappings in metric spaces. The statements on the solvability of systems of operator equations are proved for the case when the left-hand side of an equation is a value of a vector covering mapping and the right-hand side is Lipschitzian vector mapping. In the scalar case the obtained statements are equivalent to the coincide point theorems by A. V. Arutyunov. As an application, we prove a statement on the existence of $n$-fold coincidence points and obtain estimates of the points. The sufficient conditions for $n$-fold fixed points existence, including the well-known theorems on double fixed point, follow from the obtained results.