Derivative of inverse mapping and chain rule formulas for Gateaux derivatives, theorems and counter-examples

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F25%3A00605037" target="_blank" >RIV/67985840:_____/25:00605037 - isvavai.cz</a>

Výsledek na webu

<a href="https://www.heldermann.de/JCA/JCA32/JCA321/jca32015.htm" target="_blank" >https://www.heldermann.de/JCA/JCA32/JCA321/jca32015.htm</a>

DOI - Digital Object Identifier

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Jazyk výsledku

angličtina

Název v původním jazyce

Derivative of inverse mapping and chain rule formulas for Gateaux derivatives, theorems and counter-examples

Popis výsledku v původním jazyce

We investigate what happens if Fr'echet differentiability is replaced by Gateaux differentiability in several statements from differential calculus. Our main counter-example shows how the formula for derivative of inverse mapping fails when Gateaux differentiability is assumed both for the mapping and its inverse: We construct an involution $f: mathbb{R}^2 longrightarrowmathbb{R}^2$ with $f(0,0)=(0,0)$, $C^infty$-smooth off the origin and Gateaux differentiable at $(0,0)$, whose derivative $f'(0,0)$ is singular, and hence $f'(0,0),{circ},f'(0,0)$ is not the identity mapping. This mapping simultaneously provides a strong counter-example for the Chain Rule formula with Gateaux derivatives. We also show that the chain rule formula holds true for the Gateaux derivatives in an important special case of mappings between Banach spaces.

Název v anglickém jazyce

Derivative of inverse mapping and chain rule formulas for Gateaux derivatives, theorems and counter-examples

Popis výsledku anglicky

We investigate what happens if Fr'echet differentiability is replaced by Gateaux differentiability in several statements from differential calculus. Our main counter-example shows how the formula for derivative of inverse mapping fails when Gateaux differentiability is assumed both for the mapping and its inverse: We construct an involution $f: mathbb{R}^2 longrightarrowmathbb{R}^2$ with $f(0,0)=(0,0)$, $C^infty$-smooth off the origin and Gateaux differentiable at $(0,0)$, whose derivative $f'(0,0)$ is singular, and hence $f'(0,0),{circ},f'(0,0)$ is not the identity mapping. This mapping simultaneously provides a strong counter-example for the Chain Rule formula with Gateaux derivatives. We also show that the chain rule formula holds true for the Gateaux derivatives in an important special case of mappings between Banach spaces.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

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OECD FORD obor

10101 - Pure mathematics

Projekt

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Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2025

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Journal of Convex Analysis

ISSN

0944-6532

e-ISSN

0944-6532

Svazek periodika

32

Číslo periodika v rámci svazku

1

Stát vydavatele periodika

DE - Spolková republika Německo

Počet stran výsledku

28

Strana od-do

263-290

Kód UT WoS článku

001415244300014

EID výsledku v databázi Scopus

2-s2.0-105006500453