INVESTIGATION OF A MULTIPERIODIC PROBLEM FOR A FINITEHEREDITARY SYSTEM OF INTEGRO-DIFFERENTIAL EQUATIONS

Numerous problems arising in various fields of natural sciences and engineering lead to the study of integrodifferential equations. The investigation of such equations, which account for the historical behavior of processes or phenomena, originates from the pioneering works of V. Volterra, where the role of the integral term was emphasized. Various approaches have since been developed to describe hereditary effects and aftereffects in these equations. It should be noted that the existence of multiperiodic and quasiperiodic solutions for systems of Volterra-type integrodifferential equations does not always guarantee their uniqueness. In this paper, a system of partial integro-differential equations with a special differentiation operator is considered. Using the method of periodic characteristics, integral representations of the solution manifold for such systems with aftereffects are constructed. The properties of iterated kernels and resolvents are studied, and corresponding estimates are obtained. Conditions for the existence of a Greentype matrix function for the multiperiodic problem are established, along with integral representations with suitable bounds. Finally, sufficient conditions for the existence and uniqueness of a multiperiodic solution, interpreted as an integral manifold of the integro-differential system with finite hereditary effects, are derived.

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