On the asymptotic behavior of solutions of a class of selfadjoint second order linear systems (Q1382731)

Sufficient conditions are obtained so that the second-order linear system \[ y''+ P(x)y=0 \tag{*} \] admits a nontrivial solution \(y_0(x)\) such that \(\| y_0(x) \|\to 0\) as \(x\to \infty\), where \(P(x)\) is an \(n\times n\) positive definite matrix-valued function on \([0,\infty)\). Further, it is shown, under certain conditions, that every nontrivial solution to (*) tends to zero in norm at infinity. In this case, the author assumed \(P(x)= Q^2(x)\), where \(Q(x)\) is a real symmetric matrix polynomial of degree \(\geq 1\) and with a positive definite leading coefficient. The results generalize similar results on scalar equations \(y''+ p(x)y=0\).