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 mmctctcD∂∂++∂∂+∂∂=11τ, ()constcccm−=,,1 and with finite hereditary period 0>=conste by variable τ that describe hereditary phenomena. Along with the equation of zeros of the special differentiation operator cD 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,τ with periods ()ωθ, 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 e heredity period for linear systems of integro-differential equations with a special partial differential operator cD. Multiperiodic solutions obtained along characteristics 00ττcctt−+= with fixed ()00,tτ are used as an application in the theory of quasiperiodic solutions of systems of integro-differential equations.