New Kamenev-type theorems for super-linear matrix differential equations

By using Riccati transformation and the integral averaging technique, some new Kamenev-type oscillation criteria are established for the super-linear matrix differential systems X″(t)+(Xn(t)Q(t) X∗n(t))X(t)=0 and X″(t)+(X∗n(t)Q(t)Xn(t))X(t)=0,t⩾t0>0,n≥1, where Q(t) is an m×m continuous symmetric and positive definite matrix for t∈[t0,∞). The results improve and complement those given by Tomastik [E.C. Tomastik, Oscillation of nonlinear matrix differential equations of second-order, Proc. Amer. Math. Soc. 19 (1968) 1427-1431], Ahlbrandt et al. [C.D. Ahlbrandt, J. Ridenhour, R.C. Thompson, Oscillation of super-linear matrix differential equation, Proc. Amer. Math. Soc. 105 (1989) 141-148] and Ou [L.M. Ou, Atkinson's super-linear oscillation theorem for matrix dynamic equations on a time scale, J. Math. Anal. Appl. 299 (2004) 615-629], which is illustrated by an example at the end of the paper.