The functional-differential equation y′(t)=Ay(t)+By(qt)+Cy′(qt)+f(t)

An initial value problem for the functional differential equation y'(t)=Ay(t)+By(qt)+Cy'(qt)+f(t), t greater than or equal to t(0) > 0, where A, B, C are complex matrices, q is an element of(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.