Monotonicity Analysis of Generalized Discrete Fractional Proportional $h$-Differences with Applications (Journal of Nonlinear Modeling and Analysis, 2025)
Monotonicity analysis is an important aspect of fractional mathematics. In this paper, we perform a monotonicity analysis for a generalizedclass of nabla discrete fractional proportional difference on the $h\mathbb{Z}$ scale oftime. We first define the sums and differences of order $0 < α ≤ 1$ on thetime 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 thefractional proportional differences. Afterward, we introduce and prove themonotonicity results for nabla and Caputo discrete $h$-fractional proportionaldifferences. Finally, we provide two numerical examples to verify the theoretical results along with a proof for a new version of the fractional proportionaldifference of the mean value theorem on $h\mathbb{Z}$ as an application.