Abstract: Some new oscillation criteria were established for the
second order nonlinear matrix differential system $
(a(t)\X’(t))’+b(t)\X’(t)+\Q(t)f(\X(t))= 0,t\geqslant t_ 0 >0,$
where $\Q(t),$ $f’(\X(t))$ are $n \times n$ matrices with
$f’(\X(t))$ positive definite, and $a(t),$ $b(t)$ are real-valued
functions. The criteria were presented in the form of $\lim
\sup\lambda_ 1 > const $ by using a particular function
$\phi(t,s,r)$ defined as $\phi(t,s,r)=(t-s)^ \alpha (s-r)^ \beta $,
where $\alpha,\ \beta > \frac 1 2 $ are constants and $r
\geqslant t_0.$ Our results improve many known oscillation
results.

Key words: second order, matrix differential equation, oscillation, second order, matrix differential equation