  The matrix equations of the relativistic random-phase approximation (RRPA)
are derived for an effective Lagrangian characterized by density-dependent
meson-nucleon vertex functions. The explicit density dependence of the
meson-nucleon couplings introduces rearrangement terms in the residual two-body
interaction, that are essential for a quantitative description of excited
states. Illustrative calculations of the isoscalar monopole, isovector dipole
and isoscalar quadrupole response of $^{208}$Pb, are performed in the fully
self-consistent RRPA framework, based on effective interactions with a
phenomenological density dependence adjusted to nuclear matter and ground-state
properties of spherical nuclei. The comparison of the RRPA results on multipole
giant resonances with experimental data constrains the parameters that
characterize the isoscalar and isovector channel of the density-dependent
effective interactions.
