Generalized equations of state and regular universes

We found non singular solutions for universes filled with a fluid which obey a Generalized Equation of State of the form $P(\rho)=-A\rho+\gamma\rho^{\lambda}$. An emergent universe is obtained if $A=1$ and $\lambda =1/2$. If the matter source is reinterpret as that of a scalar matter field with some potential, the corresponding potential is derived. For a closed universe, an exact bounce solution is found for $A=1/3$ and the same $\lambda$. We also explore how the composition of theses universes can be interpreted in terms of known fluids. It is of interest to note that accelerated solutions previously found for the late time evolution also represent regular solutions at early times.