A global implicit function theorem and its applications to functional equations

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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

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

Abstract
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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:

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[2]	M. Cristea, A note on global implicit function theorem,, J. Inequal. Pure and Appl., 8 (2007).
[3]	D. Idczak, A. Skowron and S. Walczak, On the diffeomorphisms between Banach and Hilbert spaces,, Advanced Nonlinear Studies, 12 (2012), 89.
[4]	A. D. Ioffe and V. M. Tikhomirov, Theory of Extremum Problems,, North-Holland, (1979).
[5]	P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations,, Amer. Math. Soc., (1986).
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[7]	M. Willem, Minimax Theorems,, Birkhauser, (1996). doi: 10.1007/978-1-4612-4146-1.
[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. doi: 10.1016/j.jmaa.2005.08.072.*

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References:

[1]	A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Funct. Anal., 14 (1973), 349. doi: 10.1016/0022-1236(73)90051-7.
[2]	M. Cristea, A note on global implicit function theorem,, J. Inequal. Pure and Appl., 8 (2007).
[3]	D. Idczak, A. Skowron and S. Walczak, On the diffeomorphisms between Banach and Hilbert spaces,, Advanced Nonlinear Studies, 12 (2012), 89.
[4]	A. D. Ioffe and V. M. Tikhomirov, Theory of Extremum Problems,, North-Holland, (1979).
[5]	P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations,, Amer. Math. Soc., (1986).
[6]	W. C. Rheinboldt, Local mapping relations and global implicit function theorems,, Trans. Amer. Math. Soc., 138* (1969), 183. doi: 10.1090/S0002-9947-1969-0240644-0.*
[7]	M. Willem, Minimax Theorems,, Birkhauser, (1996). doi: 10.1007/978-1-4612-4146-1.
[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. doi: 10.1016/j.jmaa.2005.08.072.*

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