On The Asymptotic Action of the Difference Equations Class

This paper investigates the global behavior and periodicity of solutions for a broad class of higher-order nonlinear difference equations arising in discrete dynamical systems. The study aims to establish rigorous sufficient conditions ensuring the existence and uniqueness of positive solutions under appropriate initial conditions. Using advanced techniques from nonlinear analysis, the qualitative properties of the solutions are examined, including boundedness, persistence, and convergence.Particular emphasis is placed on the asymptotic stability of equilibrium points, where explicit criteria are derived to determine the conditions under which solutions converge to steady states. The periodic behavior of solutions is also analyzed, with special attention given to the existence of prime period-two solutions under suitable parameter constraints. These results reveal the complex and rich dynamical structure that such nonlinear systems can exhibit.Furthermore, the long-term behavior of solutions is studied in detail, highlighting their convergence patterns and stability characteristics over time. To support the theoretical analysis, several numerical examples are presented, illustrating the applicability and accuracy of the obtained results. These examples clearly demonstrate both the asymptotic stability and periodic dynamics predicted by the theoretical findings, thereby confirming the validity and effectiveness of the proposed approach.