Vol. 33, No. 3, 1970

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The asymptotic behaviour of solutions to linear systems of ordinary differential equations

Jack W. Macki and James Stephen Muldowney

Vol. 33 (1970), No. 3, 693–706
Abstract

This paper is concerned with the system of differential equations

x′ = A (t)x, t ∈ [0,ω)
(1)

where A(t) is an n×n matrix of locally integrable complexvalued functions on [0) and x(t) is an n-dimensional column vector. The class of matrices A(t) such that (1) has a nontrivial solution x0(t) satisfying limtω|x0(t)| = 0 is denoted by Ω0; the class of matrices A(t) such that (1) has a solution x(t) satisfying limtω|x(t)| = +is denoted by Ω. If P is a projection then ΩP0 denotes the class of matrices A(t) such that (1) has a nontrivial solution x0(t) satisfying limtω|Px0(t)| = 0. Sufficient conditions are given for A(t) Ω0,A(t) Ω and A(t) ΩP0; the result, obtained include as special cases theorems of Coppel, Hartman, and Milloux.

Mathematical Subject Classification
Primary: 34.50
Milestones
Received: 2 June 1969
Published: 1 June 1970
Authors
Jack W. Macki
James Stephen Muldowney