Oscillation theorems for self-adjoint matrix Hamiltonian systems

Kamenev type oscillation criteria for second order matrix differential systems with damping

“…Without loss of generality, we may assume that det Y (t) = 0 for t ≥ t 0 . Define W (t) by (4). As in the proof of Theorem 2.1, we deduce that (7) implies…”

“…By arguments similar to those above, we can obtain (22) and lim sup (25) is oscillatory. However, the oscillation of our system cannot be demonstrated by the criteria in [1][2][3][4].…”

“…But the system (24) has a nonoscillatory solution x(t) = exp{− sin t}. Hence, Theorems 2.1, 2.2 and Theorem 3.2 cannot be applied to our system, neither can the oscillation criteria in [1][2][3][4].…”

On the Oscillation of Self-Adjoint Matrix Hamiltonian Systems

“…Other oscillation results based on a Wintner-type criterion for (1.1) and the special system (1.4) can also be found in a recent paper of the first author [18] and some of the references cited therein.…”

Interval oscillation criteria for linear Hamiltonian systems

“…Many of these criteria involve the integral of the coefficients modelled on either the criteria due to Wintner [19] or Kamenev [9] for the scalar equation. The Hamiltonian system (1.2) has also been investigated by many authors (see [12,16,21,22,14] etc.). Most of these oscillation criteria involve the fundamental matrix Φ(t) for the linear system v = A(t)v. Such a system generally cannot be solved for v(t).…”