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Periodic solutions for nonlinear integro-differential systems with piecewise constant argument.

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This study establishes criteria for periodic solutions in nonlinear integro-differential equations with generalized arguments (DEPCAG). It confirms the existence of both periodic and unique periodic solutions using fixed-point theorems.

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Area of Science:

  • Mathematics
  • Differential Equations
  • Nonlinear Analysis

Background:

  • Nonlinear integro-differential equations with generalized arguments present unique analytical challenges.
  • Investigating periodic solutions is crucial for understanding the long-term behavior of dynamical systems.

Purpose of the Study:

  • To establish existence criteria for periodic solutions of nonlinear integro-differential equations with piecewise alternately advanced and retarded arguments (DEPCAG).
  • To analyze the critical case where the associated linear homogeneous system has nontrivial periodic solutions.

Main Methods:

  • Converting the DEPCAG into an equivalent integral equation using Green's function for periodic solutions.
  • Applying Krasnoselskii's fixed-point theorem to demonstrate the existence of a periodic solution.
  • Utilizing the contraction mapping principle to prove the existence of a unique periodic solution.

Main Results:

  • Derived criteria for the existence of periodic solutions for the studied class of DEPCAG.
  • Demonstrated the existence of at least one periodic solution.
  • Proved the existence of a unique periodic solution under specific conditions.

Conclusions:

  • The study successfully establishes conditions for the existence of periodic solutions in complex nonlinear integro-differential systems.
  • The employed methods, including fixed-point theorems, provide a robust framework for analyzing such equations.
  • The findings are validated with illustrative examples, confirming the practical applicability of the theoretical results.