Fractional ℎ ‐differences with exponential kernels and their monotonicity properties

Abstract

In this work, the nabla fractional differences of order 0<𝜇<1 with discrete exponential kernels are formulated on the time scale ℎℤ , 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 ℎℤ is proved.