Asymptotic behavior of systems of linear ordinary differential equations

Abstract

Conditions will be placed on the m × m m \times m matrices G ( t ) G(t) and G i ( t ) {G_i}(t) to assure that for any integer k = 1 , … , n k = 1, \ldots ,n , the linear differential system \[ x i ′ = G i ( t ) x i + 1 , i = 1 , … , n − 1 , x n ′ = G ( t ) x 1 , {x’_i} = {G_i}(t){x_{i + 1}},\quad i = 1, \ldots ,n - 1,\quad {x’_n} = G(t){x_1}, \] where the x i {x_i} are m × m m \times m matrices, has a solution ( x 1 , … , x n ) ({x_1}, \ldots ,{x_n}) with the property that x k ( t ) → I {x_k}(t) \to I (the identity matrix) and if k > n k > n , x i ( t ) → 0 {x_i}(t) \to 0 , i = k + 1 , … , n i = k + 1, \ldots ,n , as t → ∞ t \to \infty . Furthermore, important bounds on the x i ( t ) {x_i}(t) will be given. Some of these conditions will require that ∫ a ∞ | G | > ∞ \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 (R(t)x) = G(t)x . No selfadjointness conditions are assumed; however, the results are new even in the selfadjoint case.