Local bifurcation from characteristic values with finite multiplicity and its application to axisymmetric buckled states of a thin spherical shell

The purpose of this paper is to study bifurcation points of the equation T ( v ) = L (λ, v ) + M (λ, v ), (λ, v ) ϵ Λ × D in Banach spaces, where for any fixed λ ϵ Λ, T , L (λ,·) are linear mappings and M (λ,·) is a nonlinear mapping of higher order, M (λ,0) = 0 for all λ ϵ Λ. We assume that λ is a characteristic value of the pair ( T , L ) such that the mapping T – L ( λ ,·) is Fredholm with nullity p and index s , p > s ⩾ 0. We shall find some sufficient conditions to show that ( λ ,0) is a bifurcation point of the above equation. The results obtained will be used to consider bifurcation points of the axisymmetric buckling of a thin spherical shell subjected to a uniform compressive force consisting of a pair of coupled non‐linear ordinary differential equations of second order.