Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-28T02:44:38.487Z Has data issue: false hasContentIssue false
  • English
  • Français

Perturbation of zeros in the presence of symmetries

Part of: Variational problems in infinite-dimensional spaces

Published online by Cambridge University Press:  09 April 2009

E. N. Dancer
E. N. Dancer
Affiliation:
Department of Mathematics University of New England Armidale, N.S.W. 2351, Australia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the existence of zeros of a perturbed nonlinear operator near a zero of the unperturbed operator in the case where both operators are invariant under a symmetry group. To do this, we first correct some work of Rubinsztein on the G-homotopy groups of spheres.

MSC classification

Secondary: 58E07: Abstract bifurcation theory 57S99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 36 , Issue 1 , February 1984 , pp. 106 - 125
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Amann, H. and Zehnder, E., ‘Non-trivial solutions for a class of nonresonance problems and applications to non-linear differential equations’, Ann. Scuola Norm. Sup. Pisa 7, (1980), 539–603.Google Scholar
[2]Bierstone, E., ‘General position equivariant maps’, Trans. Amer. Math. Soc. 234 (1977), 447–466.CrossRefGoogle Scholar
[3]Bredon, G., Introduction to compact transformation groups (Academic Press, New York, 1972).Google Scholar
[4]Conley, C., ‘Isolated invariant sets and the Morse index’, CBMS regional conferences in mathematics no. 38 (Amer. Math. Soc., Providence, R.I., 1978).Google Scholar
[5]Dancer, E. N., ‘On the existence of bifurcating solutions in the presence of symmetries’, Proc. Royal Soc. Edinburgh 85 A (1980), 321–336.CrossRefGoogle Scholar
[6]Dancer, E. N., ‘An implicit function theorem with symmetries and its application to nonlinear eigenvalue problems’, Bull. Austral. Math. Soc. 21 (1980), 81–91.CrossRefGoogle Scholar
[7]Dancer, E. N., ‘On the existence of solutions of certain asymptotically homogeneous problems’, Math. Z. 177 (1981), 33–48.CrossRefGoogle Scholar
[8]Dancer, E. N., ‘Symmetries, degree, homotopy indices and asymptotically homogeneous problems’, Nonliner Analysis 6 (1982), 667–686.CrossRefGoogle Scholar
[9]Dancer, E. N., ‘The G-invariant implicit function theorem in infinite dimensions’, Proc. Royal Soc. Edinburgh 92 A (1982), 13–30.CrossRefGoogle Scholar
[10]Dieudonné, J., Treatise on analysis, Vol. IV (Academic Press, New York, 1974).Google Scholar
[11]Dold, A., Lectures on algebraic topology, (Springer, Berlin, 1970).Google Scholar
[12]Field, M., ‘Transversality in G-manifolds’, Trans. Amer. Math, Soc. 231 (1977), 429–450.Google Scholar
[13]Golubitsky, M. and Schaefer, D., ‘A discussion of symmetry and symmetry breaking’, to appear.Google Scholar
[14]Golubitsky, M. and Schaefer, D., ‘Imperfect bifurcation in the presence of symmetry’, Comm. Math. Phy. 67 (1979), 205–232.CrossRefGoogle Scholar
[15]James, I. and Segel, G., ‘On equivariant homotopy type’, Topology 17 (1978), 267–272.CrossRefGoogle Scholar
[16]Jaworawski, J., ‘Extension of G-maps and Euclidean G-retracts’, Math. Z. 146 (1976), 143–148.CrossRefGoogle Scholar
[17]Komiya, K., ‘On the equivariant homotopy of Stiefel manifolds’, Osaka J. Math. 17 (1980), 589–601.Google Scholar
[18]Lloye, N., Degree theory (Cambridge University Press Cambridge, 1978).Google Scholar
[19]Rabinowitz, P., ‘A bifurcation theorem for potential operators’, Functional Analysis 25 (1977), 412–424.CrossRefGoogle Scholar
[20]Rothe, E., ‘A relation between the type numbers of a critical point ant the index of the corresponding field of gradient vectors’, Math. Nachr. 4 (1951), 12–27.Google Scholar
[21]Rubinsztein, R., ‘On the equivariant homotopy of spheres’, Dissertations Math. (Rozprawy Mat.) 134 (1976) 1–48.Google Scholar