Oscillation Results for Second Order Self-adjoint Matrix Differential Systems

on [0,∞), where Y (t), P (t) and Q(t) are n × n real continuous matrix functions on [0,∞) with P (t), Q(t) symmetric and P (t) positive definite for t ∈ [0,∞) (P (t) > 0, t ≥ 0). A solution Y (t) of (1.1) is said to be nontrivial if det Y (t) 6= 0 for at least one t ∈ [0,∞) and a nontrivial solution Y (t) of (1.1) is said to be prepared (selfconjugated) if Y ∗(t)P (t)Y ′(t)− Y ∗′(t)P (t)Y (t) ≡ 0, t ∈ [0,∞), where for any matrix A, the transpose of A is denoted by A∗. System (1.1) is said to be oscillatory on [0,∞) if the determinant of every nontrivial prepared solution vanishes on [T,∞) for each T > 0. The oscillation problem for system (1.1) and its various particular cases have been studied extensively in recent years, e.g., see [1–8] and the references therein. In 1999, Parhi and Praharaj [6] studied the oscillation of system (1.1) under the assumption P−1(t) ≥ I, where I is the n × n identity matrix. However, this assumption restricts the application of the results of [6], in particular, they cannot be applied to the case P−1(t) ≥ p−1(t)I, e.g., to a system of the form [p(t) Y ′]′ + Q(t) Y = 0, where p ∈ C([0,∞), (0,∞)). In the present paper we obtain, by employing the matrix Riccati technique and the integral averaging technique, several new oscillation criteria for system (1.1) under the assumption P−1(t) ≥ p−1(t)I. Our results extend, improve and unify a number of the existing results and enable one to handle some cases