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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
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


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].

Research Article
Copyright © Royal Society of Edinburgh 1988

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