Kamenev Type Theorems for Second Order Matrix Differential Systems

We consider the second order matrix differential systems (1) (P(t)Y1)'+ Q(t)Y = 0 and (2) Y" + Q(t)Y = 0 where Y, P , and Q are n x n real continuous matrix functions with P(t) , Q(t) symmetric and P(t) positive definite for t E [to, cc) (P(t) > 0 , t > to) . We establish sufficient conditions in order that all prepared solutions Y(t) of (1) and (2) are oscillatory. The results obtained can be regarded as generalizing well-known results of Kamenev in the scalar case. Consider the second order linear differential system where t > to and Y ( t ), P ( t ), and Q ( t ) are n x n real continuous matrix functions with P ( t ), Q ( t ) symmetric and P ( t ) positive definite for t E [ t o ,m) ( P ( t )> 0 , t 2 to ). When P ( t )I for t 2 to where I is the n x n identity matrix, we consider A solution Y ( t ) of ( 1 . 1 ) (or (1.2)) is said to be nontrivial if det Y ( t )# 0 for at least one t E [ t o , co) and a nontrivial solution Y ( t ) of (1.1) is said to be prepared if where for any matrix A , the transpose of A is denoted by A * . System ( 1 . 1 ) is said to be oscillatory on [ t o , co) in case the determinant of every nontrivial prepared solution vanishes on [ T :co) for each T > to. For the corresponding scalar equation of system (1.2), Received by the editors June 13, 199 1. 199 1 Mathematics Subject Classification. Primary 34A30, 34C10.