An initial value problem for the functional differential equation
y′(t)
=Ay(t)
+By(qt)
+Cy′(qt)
+f(t), t ≥ t0 > 0
where A, B, C are complex matrices,
q∈(0, 1), and f is a vector of continuous functions,
is
considered in this paper. Its solution is represented in terms of
the fundamental solution via the
variation-of-constants formula. For some special cases, the fundamental
solutions are
formulated as piecewise Dirichlet series. The variation-of-constants
formula is used to analysis
the asymptotic behaviour of the solutions of some scalar equations, including
one with variable
coefficients related to coherent states of the q-oscillator
algebra in quantum mechanics.