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Small solutions of a non-linear equation in Banach space for a degenerate case

Published online by Cambridge University Press:  14 February 2012

M. Shearer
M. Shearer
Affiliation:
Fluid Mechanics Research Institute, University of Essex
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Synopsis

The existence of zeroes of an operator in Banach space near a. known zero is examined, when the non-degeneracy condition of Magnus [1] is violated, and replaced by a condition which is called simple-degeneracy. Criteria are given which help determine the number and the structure of curves of zeroes of the operator, passing through the known zero.

These conditions and results are interpreted in terms of bifurcation theory. By considering several different cases, it is shown that the failure of various conditions frequently employed in the Lyapunov–Schmidt method may be overcome using the idea of simple-degeneracy.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 79 , Issue 1-2 , 1977 , pp. 35 - 49
Copyright
Copyright © Royal Society of Edinburgh 1977

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References

1Magnus, R. J.. On the local structure of the zero set of a Banach space valued mapping. J. Functional Analysis 22 (1976), 58–72.CrossRefGoogle Scholar
2Magnus, R. J.. Topological stability of the solution set of a non-linear equation. Battelle Inst. Geneva, Math. Report 85 (1974).Google Scholar
3Krasnosel'skii, V. M.. Small solutions of a class of nonlinear operator equations. Soviet Math. Dokl. 9 (1968), 579–581.Google Scholar
4Krasnosel'skii, V. M.. Investigation of the bifurcation of small eigenfunctions in the case of multidimensional degeneration. Soviet Math. Dokl. 11 (1970), 1609–1613.Google Scholar
5Krasnosel'skii, V. M.. The projection method for the study of bifurcation of the zero solution of a nonlinear operator equation with multidimensional degeneracy. Soviet Math. Dokl. 12 (1971), 967–971.Google Scholar
6McLeod, J. B. and Sattinger, D. H.. Loss of stability and bifurcation at a double eigenvalue. J. Functional Analysis 14 (1973), 62–84.CrossRefGoogle Scholar
7Sather, D.. Branching of solutions of nonlinear equations. Rocky Mountain J. Math. 3 (1973) 203–250.CrossRefGoogle Scholar
8Chow, S., Hale, J. K. and Mallet-Paret, J.. Applications of generic bifurcation, II. Arch. Rational Mech. Anal, 62 (1976), 209–236.CrossRefGoogle Scholar
9Dancer, E. N.. Bifurcation in real Banach space. Proc. London Math. Soc. 23 (1971), 699–734.CrossRefGoogle Scholar
10Schechter, M.. Principles of Functional Anatysis. (New York: Academic Press, 1971).Google Scholar