MULTIPERIODIC SOLUTIONS OF LINEAR SYSTEMS INTEGRO-DIFFERENTIAL EQUATIONS WITH Dc -OPERATOR AND  -PERIOD OF HEREDITARY

Аннотация

The article explores the questions of the initial problem and the problem of multiperiodicity solutions
of linear systems integro-differential equations with an operator of the form c m m D     c  t  c  t 1 1  ,
c   c c const  1 ,, m  and with finite hereditary period  const0 by variable  that describe hereditary
phenomena. Along with the equation of zeros of the special differentiation operator Dc are considered linear systems
of homogeneous and inhomogeneous integro-differential equations, sufficient conditions are established for the
unique solvability of the initial problems for them, both necessary and sufficient conditions of multiperiodic
existence are obtained by    ,t with periods   , of the solutions. The integral representations of multiperiodic
solutions of linear inhomogeneous systems with the uniqueness property are determined 1) in the particular case
when the corresponding homogeneous systems have exponential dichotomy and 2) in the general case when the
homogeneous systems do not have multiperiodic solutions, except for the trivial one. The article proposes a research
technique for solving problems that satisfy initial conditions and have the property of multiperiodicity with a given
 heredity period for linear systems of integro-differential equations with a special partial differential operator Dc .
Multiperiodic solutions obtained along characteristics 0 0 t  t  c  c with fixed   0 0  ,t are used as an application
in the theory of quasiperiodic solutions of systems of integro-differential equations.