A global implicit function theorem and its applications to functional equations

Dariusz Idczak

doi:10.3934/dcdsb.2014.19.2549

Article Contents

2014, Volume 19, Issue 8: 2549-2556. Doi: 10.3934/dcdsb.2014.19.2549

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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

Received: October 2013

Revised: March 2014

Published: October 2014

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Abstract

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.

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Mathematics Subject Classification: Primary: 26B10, 47J07.

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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), 349-381.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), 15 pp.
[3]	D. Idczak, A. Skowron and S. Walczak, On the diffeomorphisms between Banach and Hilbert spaces, Advanced Nonlinear Studies, 12 (2012), 89-100.
[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., Providence, 1986.
[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.
[7]	M. Willem, Minimax Theorems, Birkhauser, Boston, 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), 251-261.doi: 10.1016/j.jmaa.2005.08.072.