  This paper focuses on the fractional difference of Lyapunov functions related
to Riemann-Liouville, Caputo and Grunwald-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.
