Time scales: from Nabla calculus to Delta calculus and vice versa via duality

TL;DR

This paper demonstrates how to derive results between nabla and delta calculus on time scales using duality, with applications to calculus of variations.

Contribution

It introduces a duality-based method to connect nabla and delta calculus results and applies this to calculus of variations on time scales.

Findings

Established duality relations between nabla and delta calculus

Provided new methods to transfer results between the two calculi

Applied duality to solve problems in calculus of variations

Abstract

In this note we show how one can obtain results from the nabla calculus from results on the delta calculus and vice versa via a duality argument. We provide applications of the main results to the calculus of variations on time scales.