Existence and Uniqueness Theorem of the Solution to a Class of Nonlinear Nabla Fractional Difference System with a Time Delay

Abstract

In this paper, we investigate an existence and uniqueness theorem of the solution to a class of nonlinear nabla fractional difference system with a time delay. More precisely, observing $$\nu (t-k)^{\overline{\nu -1}}\le {t^{\bar{\nu }}}$$ , we get the evaluation of $$\nabla _{a+k}^{-\nu } ||z(t-k)||$$ , which allows us to apply the generalized Gronwall’s inequality for the solutions of nonlinear nabla fractional difference system. The theorems we establish fill the gaps in some existing papers.