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An analytical study of bifurcation problems for equations involving Fredholm mappings

Published online by Cambridge University Press:  14 November 2011

N.X. Tan
Affiliation:
Karl-Weierstraß-Institut für Mathematik der Akademie der Wissenschaften der DDR, Mohrenstraße 39, 1086 Berlin, DDR Institute of Mathematics, Hanoi Vietnam Box 631, Buu Dien BO HO, Hanoi, Vietnam

Extract

Let us consider equations in the form

where Λ is an open subset of a normed space. For any fixed λ ∊ Λ, T, L(λ,.) and M(λ,.) are mappings from the closure D0 of a neighbourhood D0 of the origin in a Banach space X into another Banach space Y with T(0) = L(λ, 0) = M(λ, 0) = 0. Let λ be a characteristic value of the pair (T, L) such that TL(λ,.) is a Fredholm mapping with nullity p and index s, p> s≧ 0. Under sufficient hypotheses on T, L and M, (λ, 0) is a bifurcation point of the above equations. Some well-known results obtained by Crandall and Rabinowitz [2], McLeod and Sattinger [5] and others will be generalised. The results in this paper are extensions of the results obtained by the author in [7].

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1988

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References

1Buchner, M., Marsden, J. and Schecter, S.. Applications of the blowing-up construction and algebraic geometry to bifurcation problems. J. Differential Equations 48 (1983), 404433.CrossRefGoogle Scholar
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4Kluge, R.. Zur Lösung eines Bifurkationsproblems für die Karmanschen Gleichungen im Fall der rechteckigen Platte. Math. Nachr. 44 (1970), 2954.CrossRefGoogle Scholar
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6Nirenberg, L.. Topics in nonlinear analysis, Courant Institute Lecture Notes (New York: Courant Institute, 1974).Google Scholar
7Tan, N. X.. An analytical approach to bifurcation problems with applications to partial differential equations. Math. Nachr. 131 (1987), 251285.CrossRefGoogle Scholar
8Weinberg, M. M. and Trenogin, W. A.. Theorie der Lösungsverzweigung bei nichtlinearen Gleichungen (Berlin: Akademie, 1973).Google Scholar
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