General bounded multiperiodic solutions of linear equation with differential operator in the direction of the main diagonal

Abstract

General bounded multiperiodic solutions of linear equation with differential operator in the direction of the main diagonal In 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.

Research of Multiperiodic Solutions of Perturbed Linear Autonomous Systems With Differentiation Operator on the Vector Field

“…The foundations of the method used in this note were laid in [1,2], which were further developed in [3][4][5][6][7][8][9][10] and applied to the study of solutions different problems in the partial differential equations [11,12]. These methods with simple modifications extend to the study solutions of problems of the differential and integro-differential equations of different types [1][2][3][4][5][6][7][8][9][10][11][12], in particular, problems on multi-frequency solutions of equations from control theory [13]. The methods of research for multiperiodic solutions are successfully combined by methods for studying solutions of boundary value problems for equations of mathematical physics.…”

Research of Multiperiodic Solutions of Perturbed Linear Autonomous Systems With Differentiation Operator on the Vector Field

“…The foundations of the method used in this note were laid in [1,2], which were further developed in [3][4][5][6][7][8][9][10][11][12][13][14] and applied to the study of solutions different problems in the partial differential equations [15,16]. These methods with simple modifications extend to the study solutions of problems of the differential and integro-differential equations of different types [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16], in particular, problems on multi-frequency solutions of equations from control theory [17]. Many oscillatory phenomena are described by systems with a differentiation operator with respect to toroidal vector fields, and new methods based on the ideas of the Fourier [18], Poincaré-Lyapunov and Hamilton-Jacobi methods [19,20] appear to establish their periodic oscillatory solutions.…”