Monotonicity Analysis of Fractional Proportional Differences

Abstract

In this work, the nabla discrete new Riemann-Liouville and Caputo fractional proportional differences of order $0<\varepsilon<1$ on the time scale $ \mathbb{Z} $ are formulated. The differences and summations of discrete fractional proportional are detected on $\mathbb{Z}$, and the fractional proportional sums associated to $ \left( ^{R} _{c} \nabla ^{\varepsilon , \rho} \chi \right)(z) $ with order $0<\varepsilon<1$ 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-1} ^{R} \nabla ^{\varepsilon , \rho} \chi )(z) > 0 $ then $\chi(z)$ is $ \varepsilon \rho \ -$increasing, and if $\chi(z)$ is strictly increasing on $ \mathbb{N}_{c} $ and $\chi(c)>0$, then ($_{c-1} ^{R} \nabla ^{\varepsilon , \rho } \chi )(z) > 0$. As an application of our findings, a new version of fractional proportional difference of the Mean Value Theorem(MVT) on $\mathbb{Z}$ is proved.