Abstract
The classical inverse/implicit function theorem revolves around solving an equation involving a differentiable function in terms of a parameter and tells us when the solution mapping associated with this equation is a differentiable function. Already in 1927 Hildebrand and Graves observed that one can put aside differentiability using instead Lipschitz continuity. Subsequently, Graves developed various extensions of this idea, most known of which are the Lyusternik-Graves theorem, where the inverse of a function is a set-valued mapping with certain Lipschitz type properties, and the Bartle-Graves theorem which establishes the existence of a continuous and calm selection of the inverse. In the last several decades more sophisticated results have been obtained by employing various concepts of regularity of mappings acting in metric spaces, mainly aiming at applications to numerical analysis and optimization. This paper presents a unified view to the inverse function theorems that originate from the works of Graves. It has a historical flavor, but not entirely, tracing the development of ideas from a personal perspective rather than surveying the literature.
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Notes
- 1.
Bob passed away September 18, 2002.
- 2.
- 3.
Michael’s theorem was not known at that time.
- 4.
Interestingly enough, Hildebrand and Graves cite the 1922 paper by Banach published in Fundamenta Mathematicae, where Banach presented his contraction mapping theorem, but they prove it independently in their Theorem 1. Apparently, the contracting mapping iteration was known to Picard and Goursat long before Banach.
- 5.
There are some partial extensions to infinite dimensions but we shall not go into that here.
- 6.
This counterexample was communicated to the author by Radek Cibulka.
- 7.
- 8.
A predecessor of that theorem was given by Lyusternik, for a statement and a comparison, see [10, Section 5D].
- 9.
A more general version of Theorem 7.12 will be presented in author’s paper Bartle-Graves theorem revisited, submitted to Set-Valued and Variational Analysis, July 2019.
- 10.
As noted by the referee of this paper, here it is sufficient to assume that \(\bar x\) is in the core of the domain of g.
References
Bartle, R.G., Graves, L.M.: Mappings between function spaces. Trans. Amer. Math. Soc. 72, 400–413 (1952)
Borwein, J.M., Dontchev, A.L.: On the Bartle-Graves theorem. Proc. Amer. Math. Soc. 131, 2553–2560 (2003)
Cibulka, R., Dontchev, A.L.: A nonsmooth Robinson’s inverse function theorem in Banach spaces. Math. Program. A 156, 257–270 (2016)
Cibulka, R., Dontchev, A.L., Veliov, V.M.: Lyusternik-Graves theorems for the sum of a Lipschitz function and a set-valued mapping. SIAM J. Control Optim. 54, 3273–3296 (2016)
Clarke, F.H.: On the inverse function theorem. Pacific Journal of Mathematics 64, 97–102 (1976)
Dmitruk, A.V., Milyutin, A.A., Osmolovskií, N.P.: Ljusternik’s theorem and the theory of the extremum. Uspekhi Mat. Nauk 35, 11–46 (1980)
Dontchev, A.L.: Implicit function theorems for generalized equations. Math. Program. A 70, 91–106 (1995)
Dontchev, A.L.: The Graves theorem revisited. J. Convex Anal. 3, 45–53 (1996)
Dontchev, A.L.: A local selection theorem for metrically regular mappings. J. Convex Anal. 11, 81–94 (2004)
Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. A view from variational analysis., 2nd edn. Springer (2014)
Graves, L.M.: Some mapping theorems. Duke Math. Journal 17, 111–114 (1950)
Hildebrand, H., Graves, L.M.: Implicit functions and their differentials in general analysis. Trans. AMS 29, 127–153 (1927)
Ioffe, A.: Variational Analysis of Regular Mappings. Theory and Applications. Springer (2017)
Izmailov, A.: Strongly regular nonsmooth generalized equations. Math. Program. Ser. A 147, 581–590 (2014)
Karush, W.: Minima of functions of several variables with inequalities as side conditions. Traces and emergence of nonlinear programming. Birkhäuser/Springer Basel AG, Basel (2014)
Páles, Z.: Inverse and implicit function theorems for nonsmooth maps in Banach spaces. J. Math. Anal. Appl. 209, 202–220 (1997)
Pourciau, B.: Analysis and optimization of Lipschitz continuous mappings. J. Opt. Theory Appl. 22, 311–351 (1977)
Robinson, S.M.: Strongly regular generalized equations. Math. of Oper. Res. 5, 43–62 (1980)
Acknowledgements
The author wishes to thank Radek Cibulka for his help when preparing this paper. This work was supported by the National Science Foundation (NSF) Grant 156229, the Austrian Science Foundation (FWF) Grant P31400-N32, and the Australian Research Council (ARC) Project DP160100854.
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Dontchev, A.L. (2019). The Inverse Function Theorems of L. M. Graves. In: Bauschke, H., Burachik, R., Luke, D. (eds) Splitting Algorithms, Modern Operator Theory, and Applications. Springer, Cham. https://doi.org/10.1007/978-3-030-25939-6_7
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