Monotonicity Results for Delta and Nabla Caputo and Riemann Fractional Differences via dual identities

TL;DR

This paper establishes monotonicity properties for various fractional difference operators, including Riemann, Caputo, delta, and nabla types, by leveraging dual identities and the Q-operator, advancing the theoretical understanding of fractional differences.

Contribution

It introduces new monotonicity results for nabla and Caputo fractional differences using dual identities, extending previous separate results for delta and nabla operators.

Findings

Monotonicity properties for nabla fractional differences proved via dual identities.

Monotonicity results for Caputo fractional differences established.

Results for right fractional difference operators using Q-operator dual identities.

Abstract

Recently, some authors have proved monotonicity results for delta and nabla fractional differences separately. In this article, we use dual identities relating delta and nabla fractional difference operators to prove shortly the monotonicity properties for the (left Riemann) nabla fractional differences using the corresponding delta type properties. Also, we proved some monotonicity properties for the Caputo fractional differences. Finally, we use the $Q-$operator dual identities to prove monotonicity results for the right fractional difference operators.