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.