Oscillatory and non oscillatory criteria for the systems of two linear first order two by two dimensional matrix ordinary differential equations

Abstract

The Riccati equation method is used to establish an oscillatory and a non oscillatory criteria for nonhomogeneous linear systems of two first-order ordinary differential equations. It is shown that the obtained oscillatory criterion is a generalization of J. S. W. Wong's oscillatory criterion.

Oscillation criteria for linear matrix Hamiltonian systems

“…Remark 2.2. Explicit oscillatory criteria for the system (2.1) (therefore for the system (2.3)) are obtained in [12].…”

Oscillatory criteria for the hamiltonian systems

“…If the system (2.2) is oscillatory then from the Sturm type comparison Theorem 3.8 of work [5] (see [5], p. 1511) it follows that for any T ≥ t 0 there exists T 1 > T such that the system (2.2) is oscillatory on [T ; T 1 ]. Due to Remark 2.1 from here and from Theorem 2.1 we immediately get:…”

“…Remark 2.3. Another oscillatory criteria for the system (2.1) are proved in [5], which are applicable to the systems (2.3) and (2.6).…”

A generalization of Sturm's comparison theorem

“…However it is possible to obtain a generalization (in some other sense) of Theorem 1.1 for the systems (1.5 j ), j = 1, 2 with the conditions (1.7). The idea of a generalization of Theorem 1.1 for the systems (1.5 j ), j = 1, 2 with the conditions (1.7) is based on the concept of so called 'null-classes' of zeroes of components φ(t), ψ(t) of solutions (φ(t), ψ(t)) of two dimensional linear systems (this concept is introduced in [2]).…”