On the Dimension of the Zero or Infinity Tending Sets for Linear Differential Equations

There are well-known conditions which guarantee that all solutions to a system of differential equations <!-- MATH $x' = A(t)x$ --> , <!-- MATH $t \in [0,\omega )$ --> , satisfy <!-- MATH ${\lim _{t \to \omega }}|x(t)| = 0$ --> . Under certain stability assumptions on the system, Hartman [2], Coppel [1] and Macki and Muldowney [4] give necessary and sufficient [sufficient] conditions that the system has at least one nontrivial solution satisfying <!-- MATH $\mathop {\lim }\limits_{t \to \omega } |x(t)| = 0[\infty ]$ --> . These results are extended by studying a sequence of matrices <!-- MATH ${A^{[k]}}(t)$ --> , <!-- MATH $k = 1, \ldots ,n$ --> , related to such that, under the same stability assumptions as before, the given system has an <!-- MATH $(n - k + 1)$ --> -dimensional zero [infinity] tending solution set if and only if [if] all nontrivial solutions of the system <!-- MATH $y' = {A^{[k]}}(t)y$ --> tend to zero [infinity].