A note on Kamenev type theorems for second order matrix differential systems

Abstract

Some oscillation criteria are given for the second order matrix differential system Y + Q ( t ) Y = 0 Y+Q(t) Y=0 , where Y Y and Q Q are n × n n\times n real continuous matrix functions with Q ( t ) Q(t) symmetric, t ∈ [ t 0 , ∞ ) t\in [t_0,\infty ) . 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 } V(t)=a(t)\{Y’(t) Y^{-1}(t) +f(t)I\} , where I I is the n × n n\times n identity matrix, f ∈ C 1 f\in C^1 is a given function on [ t 0 , ∞ ) [t_0,\infty ) and a ( t ) = exp ⁡ { − 2 ∫ t f ( s ) d s } a(t)=\exp \{-2 \int ^t f(s)\,ds\} .