General bounded multiperiodic solutions of linear equation with di˙erential operator in the direction of the main diagonal

Abstract

In this article we determine the structure of the general solution of a n-th order linear equation with di˙erential operator in the direction of the main diagonal in a space of independent variables, and with coe􀀁cients being constant on the characteristic of this operator under some condition on its eigenvalues. It is assumed that the coe􀀁cients 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 di˙erential operator in the direction of the main diagonal.