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Abstract This paper focuses on the fractional difference of Lyapunov functions related to Riemann–Liouville, Caputo and Grünwald–Letnikov definitions. A new way of building Lyapunov functions is introduced and then five inequalities are derived for each definition. With the help of the developed inequalities, the sufficient conditions can be obtained to guarantee the asymptotic stability of the nabla discrete fractional order nonlinear systems. Finally, three illustrative examples are presented to demonstrate the validity and feasibility of the proposed theoretical results. Highlights • New method to build Lyapunov functions for discrete fractional order systems. • Five inequalities on the fractional difference of Lyapunov functions. • The applicability of the proposed results for three different definitions. [ABSTRACT FROM AUTHOR]

This paper analyzes the stability of fractional-order neural networks (FNNs) without and with delay by the fractional Lyapunov direct method and the fractional Razumikhin-type theorem, respectively. S-procedure is applied to handle the nonlinear constraints to obtain a wider parameter selection of the systems. For FNNs without delay, the improved less conservative conditions of the existence and uniqueness of the equilibrium point and the global Mittag-Leffler stability are all derived in the form of linear matrix inequalities (LMIs). Moreover, an LMI-based uniform stability condition of FNNs with time delay is established, which simplifies and extends some previous work. Finally, the validity of the presented results is indicated by some numerical examples. [ABSTRACT FROM AUTHOR]