American Institute of Mathematical Sciences

Advanced
October  2014, 19(8): 2549-2556. doi: 10.3934/dcdsb.2014.19.2549

A global implicit function theorem and its applications to functional equations

Dariusz Idczak 1,

1. 

Faculty of Mathematics and Computer Science, University of Lodz, Banacha 22, 90-238 Lodz, Poland

Received  October 2013 Revised  March 2014 Published  August 2014

The main result of the paper is a global implicit function theorem. In the proof of this theorem, we use a variational approach and apply Mountain Pass Theorem. An assumption guarantying existence of an implicit function on the whole space is a Palais-Smale condition. Some applications to differential and integro-differential equations are given.
Keywords: Implicit function theorem, integro-differential equation., Mountain Pass Theorem, Palais-Smale condition, differential equation.
Mathematics Subject Classification: Primary: 26B10, 47J0.
Citation: Dariusz Idczak. A global implicit function theorem and its applications to functional equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2549-2556. doi: 10.3934/dcdsb.2014.19.2549
References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[2]

M. Cristea, A note on global implicit function theorem, J. Inequal. Pure and Appl., 8 (2007), 15 pp.  Google Scholar

[3]

D. Idczak, A. Skowron and S. Walczak, On the diffeomorphisms between Banach and Hilbert spaces, Advanced Nonlinear Studies, 12 (2012), 89-100.  Google Scholar

[4]

A. D. Ioffe and V. M. Tikhomirov, Theory of Extremum Problems, North-Holland, 1979.  Google Scholar

[5]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, Amer. Math. Soc., Providence, 1986.  Google Scholar

[6]

W. C. Rheinboldt, Local mapping relations and global implicit function theorems, Trans. Amer. Math. Soc., 138 (1969), 183-198. doi: 10.1090/S0002-9947-1969-0240644-0.  Google Scholar

[7]

M. Willem, Minimax Theorems, Birkhauser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[8]

W. Zhang and S. S. Ge, A Global Implicit Function Theorem without initial point and its applications to control of non-affine systems of high dimensions, J. Math. Anal. Appl, 313 (2006), 251-261. doi: 10.1016/j.jmaa.2005.08.072.  Google Scholar

show all references

References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[2]

M. Cristea, A note on global implicit function theorem, J. Inequal. Pure and Appl., 8 (2007), 15 pp.  Google Scholar

[3]

D. Idczak, A. Skowron and S. Walczak, On the diffeomorphisms between Banach and Hilbert spaces, Advanced Nonlinear Studies, 12 (2012), 89-100.  Google Scholar

[4]

A. D. Ioffe and V. M. Tikhomirov, Theory of Extremum Problems, North-Holland, 1979.  Google Scholar

[5]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, Amer. Math. Soc., Providence, 1986.  Google Scholar

[6]

W. C. Rheinboldt, Local mapping relations and global implicit function theorems, Trans. Amer. Math. Soc., 138 (1969), 183-198. doi: 10.1090/S0002-9947-1969-0240644-0.  Google Scholar

[7]

M. Willem, Minimax Theorems, Birkhauser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[8]

W. Zhang and S. S. Ge, A Global Implicit Function Theorem without initial point and its applications to control of non-affine systems of high dimensions, J. Math. Anal. Appl, 313 (2006), 251-261. doi: 10.1016/j.jmaa.2005.08.072.  Google Scholar

[1]

Antonio Azzollini. On a functional satisfying a weak Palais-Smale condition. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 1829-1840. doi: 10.3934/dcds.2014.34.1829
[2]

Scott Nollet, Frederico Xavier. Global inversion via the Palais-Smale condition. Discrete & Continuous Dynamical Systems, 2002, 8 (1) : 17-28. doi: 10.3934/dcds.2002.8.17
[3]

A. Azzollini. Erratum to: "On a functional satisfying a weak Palais-Smale condition". Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4987-4987. doi: 10.3934/dcds.2014.34.4987
[4]

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
[5]

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
[6]

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
[7]

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
[8]

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
[9]

Christopher Grumiau, Marco Squassina, Christophe Troestler. On the Mountain-Pass algorithm for the quasi-linear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1345-1360. doi: 10.3934/dcdsb.2013.18.1345
[10]

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
[11]

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
[12]

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
[13]

Dorota Bors. Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 29-43. doi: 10.3934/dcdsb.2018003
[14]

Ankit Kumar, Kamal Jeet, Ramesh Kumar Vats. Controllability of Hilfer fractional integro-differential equations of Sobolev-type with a nonlocal condition in a Banach space. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021016
[15]

Farid Tari. Geometric properties of the integral curves of an implicit differential equation. Discrete & Continuous Dynamical Systems, 2007, 17 (2) : 349-364. doi: 10.3934/dcds.2007.17.349
[16]

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
[17]

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
[18]

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
[19]

Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Singular integro-differential equations with applications. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021051
[20]

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

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (273)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences