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An implicit function theorem with symmetries and its application to nonlinear eigenvalue equations

Published online by Cambridge University Press:  17 April 2009

E.N. Dancer
E.N. Dancer
Affiliation:
Department of Mathematics, University of New England, Armidale, New South Wales 2351, Australia.
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Abstract

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In this paper we prove a G-invariant implicit function theorem and indicate how it can be used to improve an earlier result of the author on the bifurcation of solutions of nonlinear equations in the presence of continuous groups of symmetries. We also use our theorem to show that, under reasonable hypotheses, the method of looking for solutions in invariant subspaces yields all solutions. This can be used to answer a question raised by Sattinger [J. Math. Phys. 19 (1978), 1729]. The abstract result is also of interest because it provides a theorem which should be of use in other symmetric situations.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 21 , Issue 1 , February 1980 , pp. 81 - 91
Copyright
Copyright © Australian Mathematical Society 1980

References

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