Monotonicity Analysis of Fractional Proportional Differences

Abstract

In this work, the nabla discrete new Riemann–Liouville and Caputo fractional proportional differences of order on the time scale are formulated. The differences and summations of discrete fractional proportional are detected on , and the fractional proportional sums associated to with order are defined. The relation between nabla Riemann–Liouville and Caputo fractional proportional differences is derived. The monotonicity results for the nabla Caputo fractional proportional difference are proved; specifically, if then is increasing, and if is strictly increasing on and , then . As an application of our findings, a new version of the fractional proportional difference of the mean value theorem (MVT) on is proved.