This study delves into the origin, evolution, and practical applications of fractional difference inequalities based on recent literature. The review provides an overview of existing inequalities proposed under various definitions. Furthermore, to enhance this potent mathematical tool, a series of new inequalities have been introduced. Additionally, leveraging renowned Lyapunov functions in continuous-time domain, their discrete-time counterparts have been formulated. Moreover, several new potential Lyapunov functions have been identified. This review aims to aid readers in selecting suitable inequalities and Lyapunov functions to analyze the stability of nabla fractional order systems.

本研究根据最新文献深入探讨了分数差分不等式的起源、演变和实际应用。该审查概述了根据各种定义提出的现有不平等现象。此外，为了增强这一强大的数学工具，引入了一系列新的不等式。此外，利用连续时间域中著名的李亚普诺夫函数，制定了其离散时间对应函数。此外，还确定了几个新的潜在李亚普诺夫函数。本综述旨在帮助读者选择合适的不等式和李雅普诺夫函数来分析 nabla 分数阶系统的稳定性。