Specular differentiation in normed vector spaces: Quasi-Mean Value and Quasi-Fermat Theorems

TL;DR

This paper introduces specular differentiation as a generalization of classical derivatives in normed spaces, establishing foundational properties and key theorems like the Mean Value and Fermat's Theorems in this new context.

Contribution

It develops the theory of specular differentiation, extending classical differentiation concepts and providing new insights into subdifferential analysis of convex functions.

Findings

Established weak forms of the Mean Value and Fermat's Theorems in specular differentiation

Identified a distinguished element of the Fréchet subdifferential via specular differentiation

Generalized Gâteaux and Fréchet derivatives in normed vector spaces

Abstract

This paper introduces specular differentiation, which generalizes G\^ateaux and Fr\'echet differentiation in normed vector spaces. We investigate its fundamental theoretical properties and establish weak forms of the Mean Value Theorem and Fermat's Theorem in the specular sense. Finally, we identify a distinguished element of the Fr\'echet subdifferential of a convex function through specular differentiation.