Interval oscillation criteria for linear matrix Hamiltonian systems

Abstract

A new approach with the Riccati equation method is used to obtain a non oscillation criterion for extended quasi linear Hamiltonian systems.

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

“…(3.9) has a solution y 1 (t) on [s k , t k ) with y 1 (s 1 ) ≥ y(s 1 ) and y 1 (t) ≥ y(t), t ∈ [s k , t k ). Hence according to (2.5) the pair of functions [15] (see also [16]). An interval oscillatory criteria for unforced Eq.…”

“…Then applying Theorem 2.1 to the pair of equations (3.8) [15] (see also [16]). An interval oscillatory criteria for unforced Eq.…”

Stability criteria for systems of two first-order linear ordinary differential equations

“…This method has been used in [1,2] to obtain some comparison criteria for Eq. (1.1) in the case n = 2 (the case of Riccati equations), which were used for qualitative study of different types of equations (see e. g. [3][4][5][6][7][8][9][10][11][12][13]). In the general case Eq.…”

Oscillation criteria for linear matrix Hamiltonian systems

“…The oscillation problem for linear matrix Hamiltonian systems is that of finding explicit conditions on the coefficients of the system, providing its oscillation. This is an important problem of qualitative theory of differential equations and many works are devoted to it (see e.g., [1,3,5,6,[13][14][15][16][17][18] and cited works therein). Among them notice the following result, obtained by S. Kumari and S.Umamaheswaram Theorem 1.1 ([11, Theorem 2.1]).…”

“…Traditionally the oscillation problem for the system (1.1) was studied under the restriction that the coefficient B(t) of the system (1.1) is positive definite (a condition, which is essential from the point of view if used methods). New approaches in the papers [5] and [6] allowed to obtain oscillation criteria in the direction of weakening (braking) the positive definiteness of B(t). In this paper we continue to study the oscillation problem for the system (1.1) in the mentioned direction.…”