Rabia Ilyas Butt, Thabet Abdeljawad, and Mujeeb Ur Rehman  . Discrete Version of Gronwall-Bellman Type Inequality and Its Application in Stability Analysis. Dynamic Systems and Applications 30 (2021) No.3,  449-459

https://doi.org/10.46719/dsa202130310

ABSTRACT.
In this article, we develop a new discrete version of Gronwall-Bellman type inequality. Then, using the newly developed  inequality to discuss Ulam-Hyers stability of a Caputo nabla fractional difference system. An example is provided to illustrate the theoretical results. Our analysis is applied on an isolated  time scale equivalent  to $h\mathbb{Z}$.

AMS (MOS) Subject Classification. 39A30.

Key Words. Gronwall-Bellman inequality, Caputo nabla fractional difference system, Existence, Uniqueness, Ulam-Hyers stability