An analysis of exponential kernel fractional difference operator for delta positivity

Abstract

Positivity analysis for a fractional difference operator including an exponential formula in its kernel has been examined. A composition of two fractional difference operators of order ( ν , μ ) \left(\nu ,\mu ) in the sense of Liouville–Caputo type operators has been analysed in cases when ν ≠ μ \nu \ne \mu and ν = μ \nu =\mu . Due to the kernel of the fractional difference operator being convergent, there has been a restriction in the domain of the solution. Incidentally, a negative lower bounded condition has been carried out through analysing the positivity results. For a better understanding, an increasing function has been considered as a test for the main results.