  A time-dependent density functional theory (TDDFT) for a quantum many-body
system on a lattice is formulated rigorously. We prove the uniqueness of the
density-to-potential mapping and demonstrate that a given density is
$v$-representable if the initial many-body state and the density satisfy
certain well defined conditions. In particular, we show that for a system
evolving from its ground state any density with a continuous second time
derivative is $v$-representable and therefore the lattice TDDFT is guaranteed
to exist. The TDDFT existence and uniqueness theorem is valid for any connected
lattice, independently of its size, geometry and/or spatial dimensionality. The
general statements of the existence theorem are illustrated on a pedagogical
exactly solvable example which displays all details and subtleties of the proof
in a transparent form. In conclusion we briefly discuss remaining open problems
and directions for a future research.
