Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T02:00:41.927Z Has data issue: false hasContentIssue false
  • English
  • Français

Simple proofs of some results in perturbed bifurcation theory

Published online by Cambridge University Press:  14 November 2011

K. J. Brown  and
R. Shivaji
K. J. Brown
Affiliation:
Department of Mathematics, Heriot–Watt University, Riccarton, Currie, Edinburgh EH14 4AS
R. Shivaji
Affiliation:
Department of Mathematics, Heriot–Watt University, Riccarton, Currie, Edinburgh EH14 4AS
Get access Rights & Permissions [Opens in a new window]

Synopsis

In this paper we discuss the existence and multiplicity of solutions to some perturbed bifurcation problems. By using sub and supersolution techniques along with an anti-maximum principle, simple proofs of some “well known” local results of perturbed bifurcation theory are obtained. The existence of global continua of solutions is proved by using degree theory arguments.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 93 , Issue 1-2 , 1982 , pp. 71 - 82
Copyright
Copyright © Royal Society of Edinburgh 1982

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

1Amann, H.. Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18 (1976), 620–709.CrossRefGoogle Scholar
2Clément, P. and Peletier, L. A.. An anti-maximum principle for second order elliptic operators. J. Differential Equations 34 (1979), 218–229.Google Scholar
3Keener, J. P.. Perturbed bifurcation theory at multiple eigenvalues. Arch. Rational Mech. Anal 56 (1974), 348–366.Google Scholar
4Keener, J. P. and Keller, H. B.. Perturbed bifurcation theory Arch. Rational Mech. Anal. 50 (1973), 159–175.CrossRefGoogle Scholar
5Potier-Ferry, M.. Perturbed bifurcation theory. J. Differential Equations 33 (1979), 112–146.Google Scholar
6Protter, M. H. and Weinberger, H. F.. Maximum principles in differential equations. (Englewood Cliffs, N.J.: Prentice-Hall, 1967).Google Scholar
7Rabinowitz, P. H.. Some aspects of nonlinear eigenvalue problems. Rocky Mountain J. Math. 3 (1973), 161–202.CrossRefGoogle Scholar
8Rabinowitz, P. H.. Pairs of positive solutions of nonlinear elliptic partial differential equations. Indiana Univ. Math. J. 23 (1973/1974), 173–186.Google Scholar
9Sattinger, D. H.. Topics in stability and bifurcation theory. Lecture Notes in Mathematics 309 (Berlin; Springer, 1973).Google Scholar
10Stakgold, I.. Branching of solutions of nonlinear equations. SIAM Rev. 13 (1971), 289–332.Google Scholar