The diamond-alpha Riemann integral and mean value theorems on time scales

TL;DR

This paper introduces the diamond-alpha Riemann integral on time scales and establishes fundamental mean value theorems, extending classical calculus results to this unified framework.

Contribution

It develops the diamond-alpha integral and proves key mean value theorems on time scales, bridging discrete and continuous calculus.

Findings

Established the diamond-alpha Fermat's stationary point theorem

Proved Rolle's, Lagrange's, and Cauchy's mean value theorems on time scales

Extended classical calculus results to the time scales setting

Abstract

We study diamond-alpha integrals on time scales. A diamond-alpha version of Fermat's theorem for stationary points is also proved, as well as Rolle's, Lagrange's, and Cauchy's mean value theorems on time scales.