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Name:
Miloslav Znojil
Reviewer number:
9689
Email:
znojil@ujf.cas.cz
Item's zbl-Number:
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, 569-590 (2003)
Classification:
34L30Nonlinear ordinary differential operators
Primary Classification:
58B05Homotopy and topological questions
Secondary Classification:
34B15Nonlinear boundary value problems
46T05Infinite-dimensional manifolds
Keywords:
Sturm-Liouville operator, nonlinear; infinite-dimensional manifolds; changes of variables; critical set as union of parallel hyperplanes; contractibility
Review:

Non-homogeneous and nonlinear Sturm-Liouville problem with Dirichlet
boundary conditions on half-line 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 single-hyperplane result
by Ambrosetti and Prodi [1] on inversion of a narrower set of maps
between Banach spaces is generalized in one-dimensional 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 non-linearity.
Remarks to the editors:

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