On the dimension of the zero or infinity tending sets for linear differential equations

Abstract:

There are well-known conditions which guarantee that all solutions to a system of $n$ differential equations $x’ = A(t)x$, $t \in [0,\omega )$, satisfy ${\lim _{t \to \omega }}|x(t)| = 0$. 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 $\lim \limits _{t \to \omega } |x(t)| = 0[\infty ]$. These results are extended by studying a sequence of matrices ${A^{[k]}}(t)$, $k = 1, \ldots ,n$, related to $A(t)$ such that, under the same stability assumptions as before, the given system has an $(n - k + 1)$-dimensional zero [infinity] tending solution set if and only if [if] all nontrivial solutions of the system $y’ = {A^{[k]}}(t)y$ tend to zero [infinity].