A finite-dimensional problem with several M-matrices and diagonal maximal monotone operators is studied. This problem includes variational inequalities with M-matrices as a partial case and appears, in particular, as a mesh approximation for a free boundary problem with several constraints and nonlinear relations. The existence of an unique solution for the problem is studied, as well as the convergence and geometric rate of the convergence for a class of the iterative methods, the Schwarz alternating-type methods among them. The application of the general results to a mesh scheme for a dam problem is considered. Parallel iterative methods are constructed on the basis of the domain decomposition, geometric convergence of these methods is justified.