Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. V. Babin, Finite dimensionality of the kernel and the cokernel of quasilinear elliptic mappings, Math. USSR Sb. 22 (1974), 427–454.
S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Grundlehren der mathematischen Wissenshaften, vol. 251, Springer-Verlag, New York-Heidelberg-Berlin, 1982.
M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321–340.
V. M. Eni, On the multiplicity of the characteristic values of an operator bundle, Mat. Issled 4 (1969), 32–41.
J. Esquinas, Optimal multiplicity in local bifurcation theory I: Generalized generic eigenvalues, J. Diff. Eq. 71 (1988), 206–215.
J. Esquinas and J. Lopez-Gomez, Optimal multiplicity in local bifurcation theory II: General case, J. Diff. Eq. 75 (1988), 72–92.
P. M. Fitzpatrick, Homotopy, linearization and bifurcation, Nonlinear Anal. 12 (1988), 171–184.
P. M. Fitzpatrick, J. PejsachowiczRabier, Degré topologique pour les opérateurs de Fredholm non linéaires, C. R. Acad. Sci. Paris Sér. I Math. t. 311 (1990), 711–716.
_____, Degree for proper C 2-Fredholm mappings on simply connected domains, J. Reine Angew. Math. (1992), To Appear.
P. M. Fitzpatrick and J. Pejsachowicz, An extension of the Leray-Schauder degree for fully nonlinear elliptic problems, in Nonlinear Functional Analysis, F. E. Browder, ed., Proc. Symp. Pure Math., vol. 45 (Part 1), 1986, 425–438.
_____, The fundamental group of the space of linear Fredholm operators and the global analysis of semilinear equations, Contemporary Math 72 (1988), 47–87.
_____, Local bifurcation for C 1-Fredholm maps, Proc. Amer. Math. Soc. 105 (1990), 995–1002.
_____, Nonorientability of the index bundle and several-parameter bifurcation, J. Funct. Anal. 98 (1991), 42–58.
_____, Parity and generalized multiplicity, Trans. Amer. Math. Soc. 326 (1991), 281–305.
_____, Orientation and the Leray-Schauder Theory for Fully Nonlinear Elliptic Boundary Value Problems, Mem. Amer. Math. Soc. (1992), In Press.
I. Gohberg, S. Goldberg and M. A. Kaashoek, Classes of Linear Operators Vol 1, Birkhäuser, 1990.
M. Golubitsky and V. Guilleman, Stable Mappings and Their Singularities, Springer-Verlag, New York-Heidelberg-Berlin, 1974.
E. Hopf, Abzweigung einer periodischen Lösung von einer Stationären Lösung eines Differentialsystems, Berichten der Mathematische-Physichen Klasse der Säshsischen Akademie der Wissenschaften zu Leipzig 19 (1942), [Translated into English in The Hopf Bifurcation and its Applications, Marsden and McCracken, Applied Math. Sci. Vol 19, Springer, 1976].
J. Ize, Bifurcation Theory for Fredholm Operators, Mem. Amer. Math. Soc. 174 (1976).
_____, Necessary and sufficient conditions for multiparameter bifurcation, Rocky Mountain J. of Math 18 (1988), 305–337.
N. Kopell and L. N. Howard, Bifurcations under nongeneric conditions, Advances in Math 13 (1974), 274–283.
M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, MacMillan, 1964.
M. A. Krasnosel'skii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Grundleheren der Mathematischen Wissenshaften, vol. 263, Springer-Verlag, New York-Heidelberg-Berlin, 1984.
N. H. Kuiper, The homotopy type of the unitary group of a Hilbert space, Topology 3 (1965), 19–30.
B. Laloux and J. Mawhin, Multiplicity, the Leary-Schauder formula and bifurcation, Journal of Diff. Equs. 24 (1977), 309–322.
R. J. Magnus, A generalization of multiplicity and the problem of bifurcation, Proc. London Math. Soc. 32 (1976), 251–278.
B. S. Mitjagin, The homotopy structure of the linear group of a Banach space, Uspehki Mat. Nauk. 72 (1970), 63–106.
L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Institute Lecture Notes, New York Univ., 1975.
J. Pejsachowicz, K-theoretic methods in bifurcation theory, Contemporary Math. 72 (1988), 193–205.
P. J. Rabier, Generalized Jordan chains and two bifurcation theorems of Krasnosel'skii, Nonlinear Anal. 13 (1989), 903–934.
P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487–513.
A. I. Šnirel'man, The degree of a quasi-ruled mapping and a nonlinear Hilbert problem, Math. USSR-Sb. 18 (1972), 376–396.
J. F. Toland, A Leray-Schauder degree calculation leading to nonstandard global bifurcation results, Bull. London Math Soc 15 (1983), 149–154.
D. Westreich, Bifurcation at eigenvalues of odd multiplicity, Proc. Amer. Math Soc. 41 (1973), 609–614.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1993 Springer-Verlag
About this chapter
Cite this chapter
Fitzpatrick, P. (1993). The parity as an invariant for detecting bifurcation of the zeroes of one parameter families of nonlinear Fredholm maps. In: Furi, M., Zecca, P. (eds) Topological Methods for Ordinary Differential Equations. Lecture Notes in Mathematics, vol 1537. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0085074
Download citation
DOI: https://doi.org/10.1007/BFb0085074
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-56461-4
Online ISBN: 978-3-540-47563-7
eBook Packages: Springer Book Archive