Fractional h ‐differences with exponential kernels and their monotonicity properties

In this work, the nabla fractional differences of order 0 < μ < 1 with discrete exponential kernels are formulated on the time scale h Z , where 0 < h ≤ 1 . Hence, the earlier results obtained in Adv. Differ. Equ. , 2017, (78) (2017) are generalized. The monotonicity properties of the h –Caputo‐Fabrizio (CF) fractional difference operator are concluded using its relation with the nabla h –Riemann‐Liouville (RL) fractional difference operator. It is shown that the monotonicity coefficient depends on the step h , and this dependency is explicitly derived. As an application, a fractional difference version of the mean value theorem (MVT) on h Z is proved.