  Sub-gap transport properties of a quantum dot (QD) coupled to two
superconducting and one metallic leads are studied theoretically, solving the
time-dependent equation of motion by the Laplace transform technique. We focus
on time-dependent response of the system induced by a sudden switching on the
QD-leads couplings, studying the influence of initial conditions on the
transient currents and the differential conductance. We derive analytical
expressions for measurable quantities and find that they oscillate in time with
the frequency governed by the QD-superconducting lead coupling and acquire
damping, due to relaxation driven by the normal lead. Period of these
oscillations increases with the superconducting phase difference $\phi$. In
particular, for $\phi=\pi$ the QD occupancy and the normal current evolve
monotonically (without any oscillations) to their stationary values. In such
case the induced electron pairing vanishes and the superconducting current is
completely blocked. We also analyze time-dependent development of the Andreev
bound states. We show, that the measurable conductance peaks do not appear
immediately after sudden switching of the QD coupling to external leads but it
takes some finite time-interval for the system needs create these Andreev
states. Such time-delay is mainly controlled by the QD-normal lead coupling.
