Hostname: page-component-745bb68f8f-l4dxg Total loading time: 0 Render date: 2025-02-05T14:43:57.976Z Has data issue: false hasContentIssue false
  • English
  • Français

8.—Bifurcation and Asymptotic Bifurcation for Non-compact Nonsymmetric Gradient Operators

Published online by Cambridge University Press:  14 February 2012

J. F. Toland
J. F. Toland
Affiliation:
Fluid Mechanics Research Institute, University of Essex.
Get access Rights & Permissions [Opens in a new window]

Synopsis

The first part of this paper is devoted to a study of the classical bifurcation problem in a Hilbert space, under the assumption that the operators involved are gradient operators, but not necessarily compact. Our approach to the problem was introduced by Krasnosel'skii, but here we show that his assumption about the compactness of the operators can be replaced by a much weaker Lipschitz type condition, without affecting the generality of his conclusions.

The rest of the paper is concerned with the analogous problem when the operator is knownto be asymptotically linear rather than Fréchet differentiable. Indeed, we show that this question can always be reduced to the first case, after some manipulation. After this manipulation the new operator is found to be a Fréchet differentiable gradient operator, and so we can invoke the results of the first part. This manipulation is in the spirit of that of [11] but is necessarily different.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 73 , 1975 , pp. 137 - 147
Copyright
Copyright © Royal Society of Edinburgh 1975

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Dancer, E. N., Bifurcation theory in real Banach space. Proc. London Math. Soc, 23, 699–734, 1971.CrossRefGoogle Scholar
2Rabinowitz, P. H., Some global results for nonlinear eigenvalue problems. J. Functional Analysis, 7, 487–513, 1971.CrossRefGoogle Scholar
3Stuart, C. A., Some bifurcation theory for k-set contractions. Proc. London Math. Soc, 27, 531–550, 1973.CrossRefGoogle Scholar
4Krasnosel'skii, M. A., Topological methods in the theory of nonlinear integral equations. New York: Pergamon, 1964.Google Scholar
5Vainberg, M. M., Variational methods for the study of nonlinear operators. San Francisco: Holden-Day, 1964.Google Scholar
6Nussbaum, R. D., Ph.D. thesis, Chicago Univ., 1969.Google Scholar
7Nussbaum, R. D., Estimates of the number of solutions of operator equations. Applicable Anal., 1, 183–200, 1971.CrossRefGoogle Scholar
8Browder, F. E., Nonlinear eigenvalue problems and group invariance, In Functional analysis and related fields. Berlin: Springer Verlag, 1970.Google Scholar
9Stuart, C. A., Self-adjoint square roots of positive self-adjoint bounded linear operators. Proc. Edinburgh Math. Soc, 18, 77–79, 1972.CrossRefGoogle Scholar
10Berger, M. S., A bifurcation theory for nonlinear elliptic partial differential equations and related systems, In Bifurcation theory and nonlinear eigenvalue problems (Keller, J. B. and Antman, S., Eds.). New York: Benjamin, 1969.Google Scholar
11Toland, J. F., Asymptotic linearity and nonlinear eigenvalue problems. Quart. J. Math. Oxford Ser., 24, 241–250, 1973.CrossRefGoogle Scholar
12Toland, J. F., Asymptotic linearity and a class of Sturm-Liouville Problems on the half line, Proc. Conf. Ordinary and Partial Differential Equations, Dundee 1974 (Sleeman, B. D. and Michael, I. M., Eds.). Berlin: Springer Verlag, 429–434, 1974.Google Scholar
13Böhme, R., Die Losung der Verzweigungsgleichungen fur nichtlineare Eigenwertprobleme. Math. Z., 127, 105–126, 1972.CrossRefGoogle Scholar