Derivative of inverse mapping and chain rule formulas for Gateaux derivatives, theorems and counter-examples
The result's identifiers
Result code in 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>
Result on the web
<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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Alternative languages
Result language
angličtina
Original language name
Derivative of inverse mapping and chain rule formulas for Gateaux derivatives, theorems and counter-examples
Original language description
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.
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2025
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Journal of Convex Analysis
ISSN
0944-6532
e-ISSN
0944-6532
Volume of the periodical
32
Issue of the periodical within the volume
1
Country of publishing house
DE - GERMANY
Number of pages
28
Pages from-to
263-290
UT code for WoS article
001415244300014
EID of the result in the Scopus database
2-s2.0-105006500453