Detecting multiplicity for systems of second order equations: an alternative approach

In this paper we are concerned with a system of second-order differential equations of the form x'' + A(t, x)x = 0, t ∈ [0, T], x ∈ R^N, where A(t, x) is a symmetric N × N matrix. We concentrate on an asymptotically linear situation and we prove the existence of multiple solutions to the Dirichlet problem associated to the system. Multiplicity is obtained by a comparison between the number of moments of verticality of the matrices which are the uniform limits of A(t, x) for |x| → 0 and |x| → +∞, respectively. For the proof, which is based on a generalized shooting approach, we provide a theorem on the existence of zeros of a class of N-dimensional vector fields.