American Institute of Mathematical Sciences

Advanced
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

[1]

Giuseppe Maria Coclite, Mario Michele Coclite. Positive solutions of an integro-differential equation in all space with singular nonlinear term. Discrete & Continuous Dynamical Systems, 2008, 22 (4) : 885-907. doi: 10.3934/dcds.2008.22.885
[2]

Shihchung Chiang. Numerical optimal unbounded control with a singular integro-differential equation as a constraint. Conference Publications, 2013, 2013 (special) : 129-137. doi: 10.3934/proc.2013.2013.129
[3]

Frederic Abergel, Remi Tachet. A nonlinear partial integro-differential equation from mathematical finance. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 907-917. doi: 10.3934/dcds.2010.27.907
[4]

Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Singular integro-differential equations with applications. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021051
[5]

Miloud Moussai. Application of the bernstein polynomials for solving the nonlinear fractional type Volterra integro-differential equation with caputo fractional derivatives. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021021
[6]

Michel Chipot, Senoussi Guesmia. On a class of integro-differential problems. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1249-1262. doi: 10.3934/cpaa.2010.9.1249
[7]

Tonny Paul, A. Anguraj. Existence and uniqueness of nonlinear impulsive integro-differential equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1191-1198. doi: 10.3934/dcdsb.2006.6.1191
[8]

Sertan Alkan. A new solution method for nonlinear fractional integro-differential equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1065-1077. doi: 10.3934/dcdss.2015.8.1065
[9]

Walter Allegretto, John R. Cannon, Yanping Lin. A parabolic integro-differential equation arising from thermoelastic contact. Discrete & Continuous Dynamical Systems, 1997, 3 (2) : 217-234. doi: 10.3934/dcds.1997.3.217
[10]

Narcisa Apreutesei, Nikolai Bessonov, Vitaly Volpert, Vitali Vougalter. Spatial structures and generalized travelling waves for an integro-differential equation. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 537-557. doi: 10.3934/dcdsb.2010.13.537
[11]

Samir K. Bhowmik, Dugald B. Duncan, Michael Grinfeld, Gabriel J. Lord. Finite to infinite steady state solutions, bifurcations of an integro-differential equation. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 57-71. doi: 10.3934/dcdsb.2011.16.57
[12]

Olivier Bonnefon, Jérôme Coville, Jimmy Garnier, Lionel Roques. Inside dynamics of solutions of integro-differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3057-3085. doi: 10.3934/dcdsb.2014.19.3057
[13]

Nestor Guillen, Russell W. Schwab. Neumann homogenization via integro-differential operators. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3677-3703. doi: 10.3934/dcds.2016.36.3677
[14]

Paola Loreti, Daniela Sforza. Observability of $N$-dimensional integro-differential systems. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 745-757. doi: 10.3934/dcdss.2016026
[15]

Elena Bonetti, Elisabetta Rocca. Global existence and long-time behaviour for a singular integro-differential phase-field system. Communications on Pure & Applied Analysis, 2007, 6 (2) : 367-387. doi: 10.3934/cpaa.2007.6.367
[16]

Hermann Brunner. The numerical solution of weakly singular Volterra functional integro-differential equations with variable delays. Communications on Pure & Applied Analysis, 2006, 5 (2) : 261-276. doi: 10.3934/cpaa.2006.5.261
[17]

Xinjie Dai, Aiguo Xiao, Weiping Bu. Stochastic fractional integro-differential equations with weakly singular kernels: Well-posedness and Euler–Maruyama approximation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021225
[18]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107
[19]

Jean-Michel Roquejoffre, Juan-Luis Vázquez. Ignition and propagation in an integro-differential model for spherical flames. Discrete & Continuous Dynamical Systems - B, 2002, 2 (3) : 379-387. doi: 10.3934/dcdsb.2002.2.379
[20]

Tomás Caraballo, P.E. Kloeden. Non-autonomous attractors for integro-differential evolution equations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 17-36. doi: 10.3934/dcdss.2009.2.17

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (203)
  • HTML views (141)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy