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January  2018, 23(1): 173-180. doi: 10.3934/dcdsb.2018011

A global inversion theorem for functions with singular points

Piotr Fijałkowski

Podhalańska Państwowa Wyższa Szkoła Zawodowa w Nowym Targu, ul. Kokoszków 71, 34-400 Nowy Targ, Poland

Received  July 2016 Revised  May 2017 Published  January 2018

In this paper, we consider a certain theorem on the global invertibility of a $C^{2n+1}$-mapping between Banach spaces with a singular point in which the derivatives of order up to $2n$ vanish. The theorem is illustrated by several applications.

Keywords: Inversion, differential map, singular point, nonlinear equation, integro-differential operator.
Mathematics Subject Classification: Primary: 46B03; Secondary: 47G20, 47J05.
Citation: Piotr Fijałkowski. A global inversion theorem for functions with singular points. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 173-180. doi: 10.3934/dcdsb.2018011
References:
[1]

P. Fijałkowski, Local inversion theorem for singular points, Nonlinear Anal., 54 (2003), 341-349.  doi: 10.1016/S0362-546X(03)00066-X.  Google Scholar

[2]

P. Fijałkowski, On a Certain Class of Locally Invertible Mapping and Their Applications, Wydawnictwo Uniwersytetu Lódzkiego, Lódź, 2003. Google Scholar

[3]

M. Galewski and M. Rădulescu, On a global implicit function theorem for locally Lipschitz maps via nonsmooth critical point theory, preprint, arXiv: 1704.04280. Google Scholar

[4]

O. Gutú, On global inverse and implicit functions, preprint, arXiv: 1508.07028. Google Scholar

[5]

J. Hadamard, Sur les transformations ponctuelles, Bull. Soc. Math. France, 34 (1906), 71-84.   Google Scholar

[6]

L. Hörmander, The Analysis of Linear Partial Differential Operators, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983. Google Scholar

[7]

D. Idczak, A. Skowron and S. Walczak, On the diffeomorphisms between Banach and Hilbert spaces, Adv. Nonlinear Stud., 12 (2012), 89-100.  doi: 10.1515/ans-2012-0105.  Google Scholar

[8]

G. Katriel, Mountain pass theorems and global homeomorphism theorems, Annales de l'I. H. P., 11 (1994), 189-209.  doi: 10.1016/S0294-1449(16)30191-3.  Google Scholar

[9]

R. Plastock, Homeomorphisms between Banach spaces, Trans. Amer. Math. Soc., 200 (1974), 169-183.  doi: 10.1090/S0002-9947-1974-0356122-6.  Google Scholar

[10]

M. Rădulescu and S. Rădulescu, Global inversion theorems and applications to differential equations, Nonlinear Anal., 4 (1980), 951-965.  doi: 10.1016/0362-546X(80)90007-3.  Google Scholar

[11]

M. Rădulescu and S. Rădulescu, An application of Hadamard-Levy's theorem to a scalar initial value problem, Proc. Amer. Math. Soc., 106 (1989), 139-143.  doi: 10.2307/2047385.  Google Scholar

[12]

G. Zampieri, Diffeomorphisms with Banach space domains, Nonlinear Anal., 19 (1992), 923-932.  doi: 10.1016/0362-546X(92)90104-M.  Google Scholar

show all references

References:
[1]

P. Fijałkowski, Local inversion theorem for singular points, Nonlinear Anal., 54 (2003), 341-349.  doi: 10.1016/S0362-546X(03)00066-X.  Google Scholar

[2]

P. Fijałkowski, On a Certain Class of Locally Invertible Mapping and Their Applications, Wydawnictwo Uniwersytetu Lódzkiego, Lódź, 2003. Google Scholar

[3]

M. Galewski and M. Rădulescu, On a global implicit function theorem for locally Lipschitz maps via nonsmooth critical point theory, preprint, arXiv: 1704.04280. Google Scholar

[4]

O. Gutú, On global inverse and implicit functions, preprint, arXiv: 1508.07028. Google Scholar

[5]

J. Hadamard, Sur les transformations ponctuelles, Bull. Soc. Math. France, 34 (1906), 71-84.   Google Scholar

[6]

L. Hörmander, The Analysis of Linear Partial Differential Operators, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983. Google Scholar

[7]

D. Idczak, A. Skowron and S. Walczak, On the diffeomorphisms between Banach and Hilbert spaces, Adv. Nonlinear Stud., 12 (2012), 89-100.  doi: 10.1515/ans-2012-0105.  Google Scholar

[8]

G. Katriel, Mountain pass theorems and global homeomorphism theorems, Annales de l'I. H. P., 11 (1994), 189-209.  doi: 10.1016/S0294-1449(16)30191-3.  Google Scholar

[9]

R. Plastock, Homeomorphisms between Banach spaces, Trans. Amer. Math. Soc., 200 (1974), 169-183.  doi: 10.1090/S0002-9947-1974-0356122-6.  Google Scholar

[10]

M. Rădulescu and S. Rădulescu, Global inversion theorems and applications to differential equations, Nonlinear Anal., 4 (1980), 951-965.  doi: 10.1016/0362-546X(80)90007-3.  Google Scholar

[11]

M. Rădulescu and S. Rădulescu, An application of Hadamard-Levy's theorem to a scalar initial value problem, Proc. Amer. Math. Soc., 106 (1989), 139-143.  doi: 10.2307/2047385.  Google Scholar

[12]

G. Zampieri, Diffeomorphisms with Banach space domains, Nonlinear Anal., 19 (1992), 923-932.  doi: 10.1016/0362-546X(92)90104-M.  Google Scholar

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