TY - JOUR
T1 - Monotonicity Analysis of Generalized Discrete Fractional Proportional $h$-Differences with Applications
AU - Qarariyah , Ammar
AU - Suwan , Iyad
AU - Abusaa , Muayad
AU - Abdeljawad , Thabet
JO - Journal of Nonlinear Modeling and Analysis
VL - 5
SP - 1704
EP - 1726
PY - 2025
DA - 2025/09
SN - 7
DO - http://doi.org/10.12150/jnma.2025.1704
UR - https://global-sci.org/intro/article_detail/jnma/24385.html
KW - Monotonicity analysis, $h$-fractional proportional difference, Caputo fractional proportional difference, fractional proportional Mean Value Theorem(MVT).
AB - <p style="text-align: justify;">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 $0 &lt; α ≤ 1$ 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.</p>