  In this paper, we consider the gravitational collapse of generalized Vaidya
space-time when the matter satisfies the equation of the state either $P=0$ or
$P=-\alpha \rho$, where $0 < \alpha < 1$. We show that in the case when type I
of matter field is dust, then the apparent horizon will never appear, but there
is now a family of null radial future-directed geodesics which terminated at
the central singularity in the past. Also, we show that in the case of negative
pressure, the result of the gravitational collapse might be the naked
singularity and the apparent horizon appears and in very short time disappears
again. In the case of the negative pressure, we show that the result of the
gravitational collapse might be the eternal naked singularity.
