Multiplicity results for a class of asymmetric weakly coupled systems of second-order ordinary differential equations

Abstract

We prove the existence and multiplicity of solutions to a two-point boundary value problem associated to a weakly coupled system of asymmetric second-order equations. Applying a classical change of variables, we transform the initial problem into an equivalent problem whose solutions can be characterized by their nodal properties. The proof is developed in the framework of the shooting methods and it is based on some estimates on the rotation numbers associated to each component of the solutions to the equivalent system.

Large-Amplitude Periodic Oscillations in Simple and Complex Mechanical Systems: Outgrowths from Nonlinear Analysis

“…As pointed out in [29], this result is to be regarded as a first step in a considerably more ambitious goal. Instead of a near-diagonal matrix, we would hope at first to be able to replace the matrix in (2.6) with a general n × n matrix, and make a connection between the eigenvalues of that matrix, the eigenvalues of the differential operator, and the multiplicity of the solutions.…”

“…So it is somewhat surprising that there has been little progress in generalising many of the later results to systems of elliptic partial differential equations or even ordinary differential equations. Recently, with F. Dalbono, I began the study of this area, [29]. We consider the system…”