Comparison theorem for conjugate points of systems of $n$th order nonselfadjoint differential equations

Abstract:

A comparison theorem for conjugate points of the two systems of linear differential equations ${x^{(n)}} - {( - 1)^{n - k}}p(t)x = 0$ and ${y^{(n)}} - {( - 1)^{n - k}}q(t)y = 0$, where $p(t)$ and $q(t)$ are $m \times m$ matrices of continuous functions, is given. It is assumed that $q(t)$ is positive with respect to a certain cone but no positivity conditions of any kind are imposed on $p(t)$. No selfadjointness conditions are assumed; however, the results are new even in the selfadjoint case.