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Name:  
Miloslav Znojil  
Reviewer number:  
9689  
Email:  
znojil@ujf.cas.cz  
Item's zblNumber:  
DE 019 028 614  
Author(s):  
Burghelea, Dan; Saldanha, Nucolau C.; Tomei, Carlos:  
Shorttitle:  
Infinite dimensional topology and structure of the critical set of nonlinear SL operators  
Source:  
J. Differ. Equatioins 188, No. 2, 569590 (2003)  
Classification:  
 
Primary Classification:  
 
Secondary Classification:  
 
Keywords:  
SturmLiouville operator, nonlinear; infinitedimensional manifolds; changes of variables; critical set as union of parallel hyperplanes; contractibility  
Review:  
Nonhomogeneous and nonlinear SturmLiouville problem with Dirichlet boundary conditions on halfline is considered. Its differential operator F is assumed generic (``tamed" by appropriate constraints), possessing a critical set C defined as a subset of the (Sobolev) domain of F where the ``differential" Fredholm operator DF has zero eigenvalue. Authors show that there exists a diffeomorphism in the domain of F which maps C into a union of isolated parallel hyperplanes (in this way, the thirty years singlehyperplane result by Ambrosetti and Prodi [1] on inversion of a narrower set of maps between Banach spaces is generalized in onedimensional case). In the proof they show, first, that each connected component of C is contractible (has trivial homotopy group  to show this, authors need oscillation theorems and stay within one dimension, therefore) and, second, that one may replace homotopy equivalences by diffeomorphisms. In this way, they do not need to assume the convexity or particular asymptotics for their nonlinearity.  
Remarks to the editors:  