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On the existence of bifurcating solutions in the presence of symmetries

Published online by Cambridge University Press:  14 November 2011

E. N. Dancer
Affiliation:
Department of Mathematics, University of New England, Armidale, N.S.W., Australia

Synopsis

We study bifurcation problems in the presence of continuous groups of symmetries and obtain theorems on the existence and uniqueness of solutions. We also briefly consider some applications.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1980

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References

1Abraham, R. and Robbin, J.. Transversal mappings and flows (New York: Benjamin, 1967).Google Scholar
2Bredon, G.. Introduction to compact transformation groups (New York: Academic, 1972).Google Scholar
3Cushing, J.. Periodic solutions of Volterra's population equation with hereditary effects. SIAM J. Appl. Math. 31 (1976), 251261.CrossRefGoogle Scholar
4Dancer, E. N.. Bifurcation theory in real Banach space. Proc. London Math. Soc. 23 (1971), 699734.CrossRefGoogle Scholar
5Dancer, E. N.. Bifurcation theory for analytic operators. Proc. London Math. Soc. 26 (1973), 359384.CrossRefGoogle Scholar
6Dancer, E. N.. Global structure of the solutions of nonlinear real analytic eigenvalue problems. Proc. London Math. Soc. 27 (1973), 747765.CrossRefGoogle Scholar
7Dancer, E. N.. Bifurcation theory in Banach spaces (Cambridge Univ. Ph.D. thesis, 1972).Google Scholar
8Dancer, E. N.. Some remarks on bifurcation in real Banach space, unpublished manuscript.Google Scholar
9Dieudonné, J.. Treatise on Analysis, Vol. IV (New York: Academic, 1974).Google Scholar
10Ize, J.. Bifurcation theory for Freholm operators. Mem. Amer. Math. Soc. 174 (1976).Google Scholar
11Karasnosel'skii, M. A.. Topological methods in the theory of nonlinear integral equations (New York: Pergamon, 1964).Google Scholar
12Krasnosel'skii, V. M.. An investigation of the bifurcation of small eigenfunctions in the case of multidimensional degeneracy. Soviet Math. Dokl. 11 (1970), 16091613.Google Scholar
13Loginov, B. V. and Trenogin, V. A.. The use of group properties to determine multi-parameter families of solutions of nonlinear equations. Math. USSR-Sb. 14 (1971), 438452.CrossRefGoogle Scholar
14Milnor, J.. Topology from a differentiable viewpoint (Charlottesville: Virginia Univ. Press, 1965).Google Scholar
15Nussbaum, R.. Some generalizations of the Borsuk-Ulam theorem. Proc. London Math. Soc. 35 (1977), 135158.Google Scholar
16Ruckert, H.. Losungverzweigung bei einer Klasse nichtlinearer Gleichungen. Math Z. 148 (1976), 245266.CrossRefGoogle Scholar
17Sather, D.. Branching of solutions of nonlinear equations. Rocky Mountain J. Math. 3 (1973), 203250.CrossRefGoogle Scholar
18Sattinger, D.. Transformation groups and bifurcation at multiple eigenvalues. Bull. Amer. Bath. Soc. 79 (1973), 709711.CrossRefGoogle Scholar
19Sattinger, D.. Group representation theory and branch points of nonlinear functional equations. SIAM J. Math. Anal. 8 (1977), 179201.CrossRefGoogle Scholar
20Sattinger, D.. Group representation theory, Bifurcation theory and pattern formulation. J. Functional Anal. 28 (1978), 58101.CrossRefGoogle Scholar
21Spanier, E.. Algebraic topology (New York: McGraw-Hill, 1966).Google Scholar
Komiya, K.. A necessary and sufficient condition for the existence of non-singular G-vector fields on G-manifolds. Osaka J. Math. 13 (1976), 537546.Google Scholar