The Abel equation in the asymptotic theory of functional differential equations

The paper discusses the asymptotic properties of solutions of the matrix delay differential equation x(t) = x(τ(t)) + Bx(t), t ε [t0, ∞) with complex constant matrices A, B and an unbounded lag. We give conditions under which the asymptotic behaviour of all solutions x of this equation can be related to the behaviour of a solution if of the Abel equation ψ(τ(t)) = ψ(t) - 1.Moreover, we discuss some properties of solutions of the Abel equation and describe the form of positive solutions of the auxiliary system of functional equations. The Abel equation, functional differential equation, asymptotic behaviour of solutions, transformation. © Birkhäuser Verlag, 2002.