Asymptotic Behavior of Systems of Linear Ordinary Differential Equations

Abstract

Conditions will be placed on the $m \times m$ matrices $G(t)$ and ${G_i}(t)$ to assure that for any integer $k = 1, \ldots ,n$, the linear differential system \[ {x’_i} = {G_i}(t){x_{i + 1}},\quad i = 1, \ldots ,n - 1,\quad {x’_n} = G(t){x_1},\] where the ${x_i}$ are $m \times m$ matrices, has a solution $({x_1}, \ldots ,{x_n})$ with the property that ${x_k}(t) \to I$ (the identity matrix) and if $k < n$, ${x_i}(t) \to 0$, $i = k + 1, \ldots ,n$, as $t \to \infty$. Furthermore, important bounds on the ${x_i}(t)$ will be given. Some of these conditions will require that $\int _a^\infty {\left | G \right | < \infty }$ while others will not. Corollaries will be given for special cases such as $(R(t)x'')'' = G(t)x$. No selfadjointness conditions are assumed; however, the results are new even in the selfadjoint case.