Bounded and multiperiodic solutions of the system of partial integro-differential equations

Abstract

The system of partial integro - differential equations with an operator of differentiation with respect to directions of vector field is considered. The considering integro - differential equation does not contain space variables. The matricant is constructed that satisfies the linear matrix equation and some of its properties and estimates are obtained that are related to multiperiodicity in time variables. An integral representation of the multiperiodic solution of this integro - differential system through the resolvent of the resolving kernel is given, recurrent relations are obtained for finding them. Some properties of iterated kernels and resolvent are established, corresponding estimates are found. On the basis of the necessary and sufficient condition of periodicity, multiperiodic solutions of a linear integro - differential equation and additional properties of solutions are found. Sufficient conditions for the existence of a bounded and unique multiperiodic solution on all independent variables of the characteristics system of integro - differential equations with a differentiation operator are established.