Periodic Solutions of a System of Nonlinear Integro-Differential Equations with a Product of Two Nonlinear Functions and Maxima
Ключевые слова:
Periodic solutions, integro-differential equations, product of two nonlinear functions, method of successive approximations, contraction mapping, existence, uniquenessАннотация
This paper investigates periodic solutions of a nonlinear fractional-order integrodifferential system involving the Gerasimov–Caputo operator, a product of two nonlinear functions, and maxima-type nonlinearities. The considered model combines hereditary effects, nonlinear memory interactions, and delayed maximal deviations, which significantly complicate the
qualitative analysis of the system.
The problem is studied under periodic boundary conditions in a Banach space framework. By
applying fractional integration techniques, the original nonlinear fractional integro-differential
system is reduced to an equivalent nonlinear functional-integral equation with maxima.
A constructive analytical approach based on the method of successive approximations and the
contraction mapping principle is developed. Explicit sufficient conditions ensuring existence
and uniqueness of periodic solutions are obtained. In addition, iterative schemes allowing the
practical computation of periodic solutions are constructed.
The existence problem for periodic solutions is further reduced to the investigation of zeros of a
nonlinear operator field generated by the associated integral system. Using homotopy arguments
and topological index methods, sufficient conditions for the existence of periodic solutions are
established.
The obtained results extend several known solvability results for nonlinear fractional dynamical
systems with delays and maxima and provide a constructive framework for studying periodic
processes with memory and nonlocal interactions.
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