Fractional difference inequalities with their implications to the stability analysis of nabla fractional order systems

This paper develops several beautiful fractional difference inequalities and fractional sum inequalities under Grünwald–Letnikov definition. To improve the practicability, two fractional difference inequalities are extended to the Caputo and Riemann–Liouville definitions. The newly developed inequalities play a key role to establish the bridge between the fractional difference of Lyapunov function and the fractional difference of system pseudo state. From this, 2 simple and effective stability criteria are constructed for nonlinear nabla fractional order systems. To highlight the validity and feasibility of our findings, three illustrative examples are presented at last.