Derivative of Inverse Mapping and Chain Rule Formulas for Gateaux Derivatives, Theorems and Counter-Examples

Marián Fabian; Jan Kolář

doi:10.68381/jca32015

Abstract

We investigate what happens if Fréchet 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 \longrightarrow\mathbb{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.

Author information

Contact details are reproduced from the original publication and may be historical.

Marián Fabian

Institute of Mathematics, Czech Academy of Sciences, Praha, Czech Republic

fabian@math.cas.cz

Jan Kolář

Institute of Mathematics, Czech Academy of Sciences, Praha, Czech Republic

kolar@math.cas.cz

Keywords

Chain rule formula
inverse mapping theorem
derivative
Frechet differentiability
Gateaux differentiability
involution

Mathematics Subject Classification

26B10, 46G05, 49J50, 58C20

References

V. I. Averbukh, O. G. Smolyanov: The theory of differentiation in linear topological spaces, Russ. Math. Surv. 22 (1967) 201–258.https://doi.org/10.1070/RM1967v022n06ABEH003761 Google Scholar
H. Cartan: Calcul Différentiel. I: Calcul Différentiel dans les Espaces de Banach, Herman, Paris (1967).Google Scholar
R. Cibulka, M. Fabian, A. Y. Kruger: On semiregularity of mappings, J. Math. Analysis Appl. 473/2 (2019) 811–836.https://doi.org/10.1016/j.jmaa.2018.12.071 Google Scholar
A. L. Dontchev, R. T. Rockafellar: Implicit Functions and Solution Mappings. A View from Variational Analysis, 2nd ed., Springer Series in Operations Research and Financial Engineering, Springer, New York (2014).https://doi.org/10.1007/978-1-4939-1037-3 Google Scholar
P. Drábek, J. Milota: Methods of Nonlinear Analysis: Applications to Differential Equations, 2nd ed., Birkhäuser Advanced Texts, Birkhäuser, Basel (2013).https://doi.org/10.1007/978-3-0348-0387-8 Google Scholar
M. Fabian: Concerning a geometrical characterization of Fréchet differentiability, Časopis Pěst. Matematiky 104 (1979) 45–64.https://doi.org/10.21136/CPM.1979.117999 Google Scholar
M. Fabian: Differentiability via one sided directional derivatives, Proc. Am. Math. Soc. 82 (1981) 495–500.https://doi.org/10.1090/S0002-9939-1981-0612748-2 Google Scholar
‘gerw’ (https://math.stackexchange.com/users/58577/gerw): Composition between Frechet and Gateaux derivative, https://math.stackexchange.com/q/4529899 (version: 2022-09-13).Google Scholar
P. Hájek, M. Johanis: Smooth Analysis in Banach Spaces, De Gruyter Series in Nonlinear Analysis and Applications Vol. 19, De Gruyter, Berlin (2014).https://doi.org/10.1515/9783110258998 Google Scholar
S. Krantz, H. R. Parks: A Primer of Real Analytic Functions, Birkhäuser, Basel (2002).https://doi.org/10.1007/978-0-8176-8134-0 Google Scholar
S. Lang: Real and Functional Analysis, 3rd ed., Graduate Texts in Mathematics, Springer, New York (1993).https://doi.org/10.1007/978-1-4612-0897-6_1 Google Scholar
S. Yamamuro: Differential Calculus in Topological Linear Spaces, Lecture Notes in Mathematics Vol. 374, Springer, Berlin (1974).https://doi.org/10.1007/BFb0061580 Google Scholar
L. Zajíček: Hadamard differentiability via Gâteaux differentiability, Proc. Amer. Math. Soc. 143/1 (2015) 279–288.https://doi.org/10.1090/S0002-9939-2014-12228-3 Google Scholar

Suggested citation

Published by Heldermann Verlag, 2025. Rights now held by Banach Press LLC.

[1] V. I. Averbukh, O. G. Smolyanov: The theory of differentiation in linear topological spaces, Russ. Math. Surv. 22 (1967) 201–258.https://doi.org/10.1070/RM1967v022n06ABEH003761 Google Scholar

[2] H. Cartan: Calcul Différentiel. I: Calcul Différentiel dans les Espaces de Banach, Herman, Paris (1967).Google Scholar

[3] R. Cibulka, M. Fabian, A. Y. Kruger: On semiregularity of mappings, J. Math. Analysis Appl. 473/2 (2019) 811–836.https://doi.org/10.1016/j.jmaa.2018.12.071 Google Scholar

[4] A. L. Dontchev, R. T. Rockafellar: Implicit Functions and Solution Mappings. A View from Variational Analysis, 2nd ed., Springer Series in Operations Research and Financial Engineering, Springer, New York (2014).https://doi.org/10.1007/978-1-4939-1037-3 Google Scholar

[5] P. Drábek, J. Milota: Methods of Nonlinear Analysis: Applications to Differential Equations, 2nd ed., Birkhäuser Advanced Texts, Birkhäuser, Basel (2013).https://doi.org/10.1007/978-3-0348-0387-8 Google Scholar

[6] M. Fabian: Concerning a geometrical characterization of Fréchet differentiability, Časopis Pěst. Matematiky 104 (1979) 45–64.https://doi.org/10.21136/CPM.1979.117999 Google Scholar

[7] M. Fabian: Differentiability via one sided directional derivatives, Proc. Am. Math. Soc. 82 (1981) 495–500.https://doi.org/10.1090/S0002-9939-1981-0612748-2 Google Scholar

[8] ‘gerw’ (https://math.stackexchange.com/users/58577/gerw): Composition between Frechet and Gateaux derivative, https://math.stackexchange.com/q/4529899 (version: 2022-09-13).Google Scholar

[9] P. Hájek, M. Johanis: Smooth Analysis in Banach Spaces, De Gruyter Series in Nonlinear Analysis and Applications Vol. 19, De Gruyter, Berlin (2014).https://doi.org/10.1515/9783110258998 Google Scholar

[10] S. Krantz, H. R. Parks: A Primer of Real Analytic Functions, Birkhäuser, Basel (2002).https://doi.org/10.1007/978-0-8176-8134-0 Google Scholar

[11] S. Lang: Real and Functional Analysis, 3rd ed., Graduate Texts in Mathematics, Springer, New York (1993).https://doi.org/10.1007/978-1-4612-0897-6_1 Google Scholar

[12] S. Yamamuro: Differential Calculus in Topological Linear Spaces, Lecture Notes in Mathematics Vol. 374, Springer, Berlin (1974).https://doi.org/10.1007/BFb0061580 Google Scholar

[13] L. Zajíček: Hadamard differentiability via Gâteaux differentiability, Proc. Amer. Math. Soc. 143/1 (2015) 279–288.https://doi.org/10.1090/S0002-9939-2014-12228-3 Google Scholar