Abstract
General bounded multiperiodic solutions of linear equation with differential operator in the direction of the main diagonalIn this article we determine the structure of the general solution of a n-th order linear equation with differential operator in the direction of the main diagonal in a space of independent variables, and with coefficients being constant on the characteristic of this operator under some condition on its eigenvalues.It is assumed that the coefficients and a given vector-function have the properties of periodicity and smoothness, where periods are rationally incommensurable positive constants.First, we study the homogeneous equation that reduces to a homogeneous linear system.Moreover, on this base, in terms of eigenvalues we establish conditions of existence of solutions being periodic with respect to all independent variables (so-called multiperiodic solutions).We give an integral representation of the multiperiodic solution of nonhomogeneous equation.The conditions for existence and uniqueness of the bounded and multiperiodic solutions of the n-th order linear nonhomogeneous equation are established.It is shown that the bounded solution of the nonhomogeneous equation is periodic in all variable solutions with a variable bounded period.This is one of the specific features of the equations with differential operator in the direction of the main diagonal.