ON THE UNIQUENESS OF THE DEGREE FOR NONLINEAR FREDHOLM MAPS OF INDEX ZERO BETWEEN BANACH MANIFOLDS

TitleON THE UNIQUENESS OF THE DEGREE FOR NONLINEAR FREDHOLM MAPS OF INDEX ZERO BETWEEN BANACH MANIFOLDS
Publication TypeJournal Article
Year of Publication2011
AuthorsBENEVIERI, PIERLUIGI, FURI, MASSIMO
Secondary TitleCommunications in Applied Analysis
Volume15
Issue2
Start Page203
Pagination216
Date Published04/2011
Type of Workscientific: mathematics
ISSN1083–2564
AMS47A53, 47H11, 58Cxx.
Abstract
In some previous papers we presented a fairly simple construction of a topological degree for C1 Fredholm maps of index zero between Banach manifolds which verifies the three fundamental properties of the classical degree theory: normalization, additivity and homotopy invariance. We show here that this degree is unique. Precisely, by an axiomatic approach similar to the one due to Amann-Weiss, we prove that there exists at most one real function satisfying the above properties, and this function must be integer valued.
URLhttp://www.acadsol.eu/en/articles/15/2/6.pdf
Short TitleON THE UNIQUENESS OF THE DEGREE...
Refereed DesignationRefereed
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REFERENCES

[1] H. Amann and S.A. Weiss, On the uniqueness of the topological degree, Math. Z., 130 (1973), 39–54.
[2] P. Benevieri and M. Furi, A simple notion of orientability for Fredholm maps of index zero between Banach manifolds and degree theory, Ann. Sci. Math. Qu ́ebec, 22 (1998), 131–148.
[3] P. Benevieri and M. Furi, On the concept of orientability for Fredholm maps between real Banach manifolds, Topol. Methods Nonlinear Anal. 16 (2000), 279–306.
[4] P. Benevieri, M. Furi, M.P. Pera, M. Spadini, About the sign of oriented Fredholm operators between Banach spaces Z. Anal. Anwendungen 22, n. 3 (2003), 619–645.
[5] A. Dold, Lectures on algebraic topology, Springer- Verlag, Berlin, 1972.
[6] K.D. Elworthy and A.J. Tromba, Differential structures and Fredholm maps on Banach manifolds, in “Global Analysis”, S. S. Chern and S. Smale Eds., Proc. Symp. Pure Math., Vol. 15, 1970, 45–94.
[7] K.D. Elworthy and A.J. Tromba, Degree theory on Banach manifolds, in “Nonlinear Functional Analysis”, F. E. Browder Ed., Proc. Symp. Pure Math., Vol. 18 (Part 1), 1970, 86–94.
[8] M.P. Fitzpatrick, J. Pejsachowicz and P.J. Rabier, Orientability of Fredholm Families and Topological Degree for Orientable Nonlinear Fredholm Mappings, Journal of Functional Analysis 124 (1994), 1–39.
[9] L. Fuhrer, Theorie des Abbildungsgrades in endlichdimensionalen Raumen, Dissertation, Freie Univ. Berlin, Berlin, 1972.
[10] L. Fuhrer, Ein elementarer analytischer Beweis zur Eindeutigkeit des Abbildungsgrades im Rn , Math. Nachr. 54 (1972), 259–267.
[11] T. Kato, Perturbation Theory for Linear Operators, Grundlehren der Mathematischen Wissenschaften, 132, Springer-Verlag, Berlin, 1980.
[12] S. Smale, An infinite dimensional version of Sard’s theorem, Amer. J. Math., 87 (1965), 861–866.