Oscillation of Nonlinear Systems of Matrix Differential Equations

For systems of matrix equations of the form £/' = A(t, U, V)V, V = -B(t,U,V) it is shown here that the oscillation problem can be reduced to the corresponding problem of "associated" (in some sense) scalar equations for which there exist numerous results. Furthermore, it is also shown that many of the existing results concerning the equation (A(t)U')' + B(t,U, U')U = 0 can be considerably improved by application of the above method. This paper concerns itself with the oscillation of systems of matrix differential equations (*) U' = A(t, U, V)V, V = B(t, U, V) where U = («,-,), V = (t><¿), A = (ay), and B = (£>,,) are real » X « matrices. By F(t, U, V) = /,-,(/, U, V) we mean /« =fij(t, Un, • • • , Unnt »11, • • • , f„„). The functions 0,7, by will be assumed continuous on IXR2n, where 1= [t0, +°o). The matrices A(t, U, V),B(t, U, V) f/-1 are symmetric for every pair { U, V} such that det U?¿ 0, and, moreover, A (t, U, V) is positive definite for every tEI and every pair { U, V}. By a solution of (*) we mean here any pair { U, V} of differentiable matrices which are defined on an infinite interval Iu.vQI (depending on the particular pair) and satisfy (*) on this interval. Extending the concept of a "prepared solution" (from the linear case), we shall say that { U, V} is a prepared solution of (*) if it is a solution such that: (1) U*(t)V(t) = V*(t)U(t), tEIu.v, (R* denotes the transpose of the matrix P). It is true that in the linear case we do not "lose" much by considering only prepared solutions, for, there, every solution is a linear combination of prepared ones (Barrett [l]). Here we shall also be concerned only with the prepared solutions of (*). The system (*) is Received by the editors August 31, 1970 and, in revised form, December 4, 1970. AMS 1969 subject classifications. Primary 3442, 3445, 3490.