Abstract
In this work, the nabla discrete new Riemann-Liouville and Caputo fractional proportional differences of order 0<epsilon<1 on the time scale DOUBLE-STRUCK CAPITAL Z are formulated. The differences and summations of discrete fractional proportional are detected on DOUBLE-STRUCK CAPITAL Z, and the fractional proportional sums associated to backward difference cR chi epsilon,rho z with order 0<epsilon<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 backward difference c-1R chi epsilon,rho z>0 then chi z is epsilon rho -increasing, and if chi z is strictly increasing on Nc and chi c>0, then backward difference c-1R chi epsilon,rho z>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.