Riccati techniques and variational principles in oscillation theory for linear systems

Abstract

ABSTRACT. We consider the seond order differential system (1) Y" + Q(i)Y = 0, where Q, Y are nxn matrices with Q = Q(t) a continuous symmetric matrixvalued function, t € [a,+00).We obtain a number of sufficient conditions in order that all prepared solutions Y(t) of (1) are oscillatory. Two approaches are considered, one based on Riccati techniques and the other on variational techniques, and involve assumptions on the behavior of the eigenvalues of Q{t) (or of its integral). These results extend some well-known averaging techniques for scalar equations to system (1).

On the Oscillation of Self-Adjoint Matrix Hamiltonian Systems

“…Self-adjoint linear Hamiltonian matrix systems arise in many dynamical problems and have been studied by many authors (see, for example, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]). In this paper, we consider linear self-adjoint Hamiltonian matrix systems of the form…”

Interval oscillation criteria for self-adjoint matrix Hamiltonian systems

“…It is well known that (2.24) is oscillatory for γ > 1 4 and non-oscillatory for γ . Applying theorem 2.6 to (2.24), we see that (2.24) is oscillatory for γ > 1 4 (see example 2.14, below). This implies that our results are sharper.…”

“…So the results of [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] cannot apply to the system containing the coefficient (2.25). In fact, the system containing the coefficient (2.25) is oscillatory by theorem 2.3.…”

“…Some results have concentrated on showing that (1.1) is oscillatory by applying a positive functional (see [10,[16][17][18]). Several others have involved the eigenvalues of the integrals of A(t), B(t) and C(t) (see [1,2,4,6,13,15] This conjecture was partly proved by several authors and finally settled by Byers et al . [2].…”

“…In 1987, Butler et al . [1] showed that the system (1.4) is oscillatory when In 1993, Erbe et al . [4] proved that system (1.4) is oscillatory if lim sup…”