Comparison theorems for conjoined bases of linear Hamiltonian systems without monotonicity

Abstract

In this paper we generalize comparison results for conjoined bases $$Y(t),{{\hat{Y}}}(t)$$ of two linear Hamiltonian differential systems proved by Elyseeva (J Math Anal Appl 444:1260–1273, 2016). In our consideration we do not impose classical monotonicity assumptions such that the majorant condition $${\mathcal {H}}(t)-\hat{\mathcal {H}}(t)\ge 0$$ for their Hamiltonians $${\mathcal {H}}(t), \hat{\mathcal {H}}(t)$$ and the Legendre conditions for $${\mathcal {H}}(t),\hat{\mathcal {H}}(t)$$. Our new comparison theorems are presented in terms of the so-called oscillation numbers associated with $$Y(t), {{\hat{Y}}}(t),$$ and the transformed conjoined basis $${\hat{Z}}^{-1}(t)Y(t)$$, where $${\hat{Z}}(t)$$ is a symplectic fundamental solution matrix of the Hamiltonian system with the Hamiltonian $$\hat{\mathcal {H}}(t)$$. The consideration is based on the comparative index theory applied to the continuous case.