Caputo delta weakly fractional difference equations

Abstract

In this paper, solutions of fractional difference equations with Caputo-type delta-based fractional difference operator of order $$\mu \sim 1$$ are compared with solutions of corresponding limit difference equations with usual first-order forward difference. It is shown that the limit initial value problems differ substantially when $$\mu \rightarrow 1^-$$ and $$\mu \rightarrow 1^+$$ . To derive convergence results, Gronwall type inequalities are proved for suitable fractional sum inequalities of general noninteger order. An illustrative example is also given.