A Combination Method for Local Bifurcation from Characteristic Values with Multiplicity

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.