Solving the main cosmological puzzles with a generalized time varying vacuum energy

We study the dynamics of the FLRW flat cosmological models in which the vacuum energy density varies with time, $\Lambda(t)$. In particular, we investigate the dynamical properties of a generalized vacuum model and we find that under certain conditions the vacuum term in the radiation era varies as $\Lambda(z) \propto (1+z)^{4}$, while in the matter era we have $\Lambda(z)\propto (1+z)^{3}$ up to $z\simeq 3$ and $\Lambda(z)=\Lambda$ for $z\le 3$. The confirmation of such a behavior would be of paramount importance because it could provide a solution to the cosmic coincidence problem as well as to the fine tuning problem, without changing the well known (from the concordance $\Lambda$ cosmology) Hubble expansion.