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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]

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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

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

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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

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