Positivity analysis for the discrete delta fractional differences of the Riemann-Liouville and Liouville-Caputo types

Abstract

In this article, we investigate some new positivity and negativity results for some families of discrete delta fractional difference operators. A basic result is an identity which will prove to be a useful tool for establishing the main results. Our first main result considers the positivity and negativity of the discrete delta fractional difference operator of the Riemann-Liouville type under two main conditions. Similar results are then obtained for the discrete delta fractional difference operator of the Liouville-Caputo type. Finally, we provide a specific example in which the chosen function becomes nonincreasing on a time set.

Keywords

• discrete fractional calculus
• discrete riemann-liouville operators
• discrete liouville-caputo fractional operators
• monotonicity and positivity