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A Combination Method for Local Bifurcation from Characteristic Values with Multiplicity
Author(s) -
Tan Nguyen Xuan
Publication year - 1991
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19911540116
Subject(s) - mathematics , banach space , multiplicity (mathematics) , bifurcation theory , bifurcation , mathematical analysis , implicit function theorem , pure mathematics , fixed point , closure (psychology) , discrete mathematics , nonlinear system , physics , quantum mechanics , economics , market economy
A combination of the LIAPUNOV‐SCHMIDT procedure, the implicit function theorems and the topological degree theory is used to investigate bifurcation points of equations of the form T ( v ) = L ( λ, v ) + M ( λ, v ), ( λ, v ) ϵ A × D̄, where A is an open subset in a normed space and for every fixed λ ϵ A, T, L (λ ·) and M (λ ·) are mappings from the closure D̄ of a neighborhood D 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 T − L ( λ ,·) is a FREDHOLM mapping with nullity p and index s, p > s ≧ 0. Under suitable hypotheses on T. L and M , ( λ , 0) is a bifurcation point of the above equations. This generalizes the results of [4], [6], [8], [13] and [14] etc. An application of the obtained results to the axisymmetric buckling problem of a thin spherical shell will be given.