A Note on Kamenev Type Theorems for Second Order Matrix Differential Systems

Some oscillation criteria are given for the second order matrix differential system Y ′′ +Q(t)Y = 0, where Y and Q are n× n real continuous matrix functions with Q(t) symmetric, t ∈ [t0,∞). These results improve oscillation criteria recently discovered by Erbe, Kong and Ruan by using a generalized Riccati transformation V (t) = a(t){Y ′(t)Y −1(t) + f(t)I}, where I is the n × n identity matrix, f ∈ C1 is a given function on [t0,∞) and a(t) = exp{−2 ∫ t f(s) ds}. Consider the second order linear differential system Y ′′ +Q(t)Y = 0, t ∈ [t0,∞), (1) where Y and Q are n × n real continuous matrix functions with Q(t) symmetric. A solution Y (t) of (1) is said to be a nontrivial solution if detY (t) 6= 0 for at least one t ∈ [t0,∞), and a nontrivial solution Y (t) of (1) is said to be prepared if Y ∗(t)Y ′(t)− (Y ∗(t))′Y (t) ≡ 0, t ∈ [t0,∞), (2) where for any matrix A, the transpose of A is denoted by A∗. System (1) is said to be oscillatory on [t0,∞) in case the determinant of every nontrivial prepared solution vanishes on [T,∞) for each T > t0. For matrix system (1), many authors have given some important simple oscillation criteria (see [1], [2], [3], [6]). We particularly mention the results of Erbe, Kong and Ruan [3] who proved the following theorem. Erbe, Kong and Ruan’s Theorem. Let H(t, s) and h(t, s) be continuous on D = {(t, s) : t ≥ s ≥ t0} such that H(t, t) = 0 for t ≥ t0 and H(t, s) > 0 for t > s ≥ t0. We assume further that the partial derivative ∂ ∂sH(t, s) = Hs(t, s) is nonpositive and continuous for t ≥ s ≥ t0 and h(t, s) is defined by Hs(t, s) = −h(t, s)[H(t, s)], (t, s) ∈ D. Finally, we assume that lim t→∞ sup 1 H(t, t0) λ1 [∫ t t0 ( H(t, s)Q(s)− 1 4 h(t, s)I ) ds ] = ∞, (3) Received by the editors May 25, 1996. 1991 Mathematics Subject Classification. Primary 34C10.