Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-24T04:43:33.861Z Has data issue: false hasContentIssue false
  • English
  • Français

SOLUTION BRANCHES OF NONLINEAR EIGENVALUE PROBLEMS ON RESTRICTED DOMAINS

Part of: Equations and inequalities involving nonlinear operators

Published online by Cambridge University Press:  13 March 2020

SHANE ARORA Open the ORCID record for SHANE ARORA [Opens in a new window]
SHANE ARORA*
Affiliation:
School of Mathematics and Statistics,University of Sydney, Camperdown, NSW2006, Australia email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We extend bifurcation results of nonlinear eigenvalue problems from real Banach spaces to any neighbourhood of a given point. For points of odd multiplicity on these restricted domains, we establish that the component of solutions through the bifurcation point either is unbounded, admits an accumulation point on the boundary, or contains an even number of odd-multiplicity points. In the simple-multiplicity case, we show that branches of solutions in the directions of corresponding eigenvectors satisfy similar conditions on such restricted domains.

Keywords

nonlinear eigenvalue problembifurcation theorycompact operators

MSC classification

Primary: 47J10: Nonlinear spectral theory, nonlinear eigenvalue problems
Secondary: 47J15: Abstract bifurcation theory
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 102 , Issue 3 , December 2020 , pp. 490 - 497
Check for updates
Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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

Dancer, E. N., ‘On the structure of solutions of non-linear eigenvalue problems’, Indiana Univ. Math. J. 23 (1973), 1069–1076.CrossRefGoogle Scholar
Deimling, K., Nonlinear Functional Analysis (Springer, Berlin, 1985).CrossRefGoogle Scholar
Dugundji, J., Topology (Allyn and Bacon, Boston, 1978), reprint of the 1966 original, Allyn and Bacon Series in Advanced Mathematics.Google Scholar
Rabinowitz, P. H., ‘Some global results for nonlinear eigenvalue problems’, J. Funct. Anal. 7 (1971), 487–513.CrossRefGoogle Scholar
Turner, R. E. L., ‘Transversality in nonlinear eigenvalue problems’, in: Contributions of Nonlinear Functional Analysis, Proceedings of a Symposium by the Mathematics Research Center (ed. Zarantonello, E. H.) (Academic Press, New York, 1971), 37–68.CrossRefGoogle Scholar
Whyburn, G. T., Topological Analysis, second, revised edn, Princeton Mathematical Series, 23 (Princeton University Press, Princeton, NJ, 1964).CrossRefGoogle Scholar