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) + (X-n(t)Q(t) X*(n)(t))X(t) = 0 and X ''(t) + (X*(n)(t)Q(t)X-n(t))X(t) = 0, t >= t(0) > 0; n >= 1, where Q(t) is an m x m continuous symmetric and positive definite matrix for t is an element of [t(0); infinity). 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. (C) 2009 Elsevier Inc. All rights reserved.