Monotonicity Analysis of Generalized Discrete Fractional Proportional $h$-Differences with Applications

Abstract

Monotonicity analysis is an important aspect of fractional mathematics. In this paper, we perform a monotonicity analysis for a generalized class of nabla discrete fractional proportional difference on the $h\mathbb{Z}$ scale of time. We first define the sums and differences of order $0 < α ≤ 1$ on the time scale for a general form of nabla fractional along with Riemann-Liouville $h$-fractional proportional sums and differences. We formulate the Caputo fractional proportional differences and present the relation between them and the fractional proportional differences. Afterward, we introduce and prove the monotonicity results for nabla and Caputo discrete $h$-fractional proportional differences. Finally, we provide two numerical examples to verify the theoretical results along with a proof for a new version of the fractional proportional difference of the mean value theorem on $h\mathbb{Z}$ as an application.