Fifth-Order Max-Type Fuzzy Difference Equations: Existence, Periodicity, Boundedness, and Persistence

Max-type fuzzy difference equations constitute an emerging research area arising from the integration of fuzzy mathematics and discrete dynamical systems. By characterizing uncertainty through fuzzy numbers, these equations provide rigorous mathematical modeling tools for practical problems involving both discreteness and uncertainty. This paper systematically investigates the dynamical properties of a class of max-type fuzzy difference equations. First, fuzzy set theory is used to transform the fuzzy difference equation into a corresponding system of parametric ordinary difference equations. Then, using the iterative method, inequality techniques, and mathematical induction, an expression for the periodic solutions of the max-type ordinary difference equation is derived, from which an expression for the periodic solutions of the max-type fuzzy difference equation is further obtained. In addition, the boundedness and persistence of the solutions to the fuzzy difference equation are proved. Finally, numerical simulations are conducted in MATLAB 2024, and the results illustrate the theoretical findings. This study not only enriches the theoretical framework of fuzzy difference equations but also provides new insights and methodological support for the modeling and analysis of uncertain discrete systems.