The eternal naked singularity formation in the case of gravitational collapse of generalized Vaidya space–time

In this paper, we consider the gravitational collapse of generalized Vaidya space–time when the matter satisfies the equation of the state either [Formula: see text] or [Formula: see text], where [Formula: see text]. We show that in the case when type I of matter field is dust, then the apparent horizon will never appear but there is no a family of null radial future-directed geodesics which terminate 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.