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

Source abstract

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