  Cosmological fluids with a Generalized Equation of State (GEoS) are here
considered, whose corresponding EoS parameter $\omega$ describes a fluid with
phantom behavior, namely $\omega<-1$, but leading to universes free of
singularities at any past or future, finite time. Thus avoiding, in particular,
the Big Bang and the Big Rip singularities, the last one considered to be
typical in phantom fluid models. More specifically, such GEoS fluid cosmologies
lead to regular Little Rip universes. A remarkable new property of these
solutions is proven here, namely that they avoid the initial singularity at
early times; therefore, they are able to describe emergent universes. Solutions
of this kind had been studied previously, but only either as late time or as
early time solutions; never as solutions covering both epochs simultaneously.
Appropriate conditions are proposed here that relate the Little Rip cosmologies
with the initial regular universe, for the future and past regimes,
respectively. This is done by taking as starting point the conditions under
which a given scale factor corresponds to a Little Rip universe.
