A global inversion theorem for functions with singular points

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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

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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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