PERIODIC SOLUTIONS FOR A SYSTEM OF FUNCTIONAL INTEGRO-DIFFERENTIAL EQUATIONS WITH PRODUCT OF TWO NONLINEAR FUNCTIONS AND WITH A FUNCTION INVOLVING VARIABLE DEVIATIONS UNDER THE MAXIMUM

This paper investigates periodic solutions of a nonlinear system of fractional functional integro-differential equations involving the Gerasimov--Caputo fractional operator, a product of two nonlinear functions, and maxima-type nonlinearities with variable deviations. The considered model combines hereditary effects, nonlocal interactions, and extremal functional dependencies, which substantially complicate the qualitative analysis of the system. The original periodic boundary-value problem is transformed into an equivalent system of nonlinear functional-integral equations. Explicit estimates for the associated nonlinear operators are obtained, and a constructive iterative procedure based on the method of successive approximations is developed. Sufficient conditions guaranteeing existence and uniqueness of periodic solutions are established by combining the contraction mapping principle with detailed operator estimates in an appropriate Banach space. Furthermore, the periodic solvability problem is reduced to the investigation of zeros of an associated nonlinear vector field. Using homotopy arguments and topological index techniques, sufficient conditions for the existence of periodic solutions are derived. The obtained results provide a constructive framework for the analysis of nonlinear fractional systems with memory, delays, maxima, and nonlocal interactions.

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