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 n differential equations $x’ = A(t)x$, t ∈ [0,ω ), satisfy lim t → ω|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 t → ω |x(t)| = 0[∞ ]. These results are extended by studying a sequence of matrices A[k](t), k = 1, … ,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].