Comparison of h-Difference Fractional Operators

Stability of nonlinear h-difference systems with n fractional orders

“…Therefore the facts presented in the paper are the generalization of results given in [35]. In [27] it is shown that the Riemann-Liouville-type fractional h-difference operator and the Caputo-type fractional h-difference operator are related to each other. Moreover, the Grünwald-Letnikov-type fractional h-difference operator can be expressed by the Riemann-Liouville-type fractional h-difference operator.…”

“…Since the Grünwald-Letnikov-type h-difference operator can be expressed by the Riemann-Liouville-type fractional h-difference operator, see [27], we restrict our consideration only to the Caputo-and Riemann-Liouville-type fractional h-difference operators.…”

“…We restrict our consideration to the fractional Caputo and Riemann-Liouville h-difference systems with n orders, because the Grünwald-Letnikov-type fractional h-difference operator t0i ∆ αi h can be expressed using the Riemann-Liouville-type fractional h-difference operator t0i ∆ αi h , see [27].…”

“…If the conditions (26) and (28) hold, then the solution X(·, X 0 ) is asymptotically stable provided that (27) holds. P r o o f .…”

“…P r o o f . From Theorem 4.4 the condition (26) gives the stability of the trivial solution provided that (27) holds. Assumption (28) implies that the solution X(·, X 0 ) = X(k) is attractive.…”