Monotonicity Analysis of Fractional Proportional Differences - Iyad Suwan; Shahd Owies; Muayad Abussa; Thabet Abdeljawad (Discrete Dynamics in Nature and Society)
In this work, the nabla discrete new Riemann–Liouville and Caputo fractional proportional differences of order 0<ε<1 on the time scale ℤ are formulated. The differences and summations of discrete fractional proportional are detected on ℤ, and the fractional proportional sums associated to ∇cRχε,ρz with order 0<ε<1 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 ∇c−1Rχε,ρz>0 then χz is ερ −increasing, and if χz is strictly increasing on ℕc and χc>0, then ∇c−1Rχε,ρz>0. As an application of our findings, a new version of the fractional proportional difference of the mean value theorem (MVT) on ℤ is proved.