  We propose a large class of nonsingular cosmologies of arbitrary spatial
curvature whose cosmic history is determined by a primeval dynamical $\Lambda
(t)$-term. For all values of the curvature, the models evolve between two
extreme de Sitter phases driven by the relic time-varying vacuum energy
density. The transition from inflation to the radiation phase is universal and
points to a natural solution of the graceful exit problem regardless of the
values of the curvature parameter. The flat case recovers the scenario recently
discussed in the literature (Perico et al., Phys. Rev. D88, 063531, 2013). The
early de Sitter phase is characterized by an arbitrary energy scale $H_I$
associated to the primeval vacuum energy density. If $H_I$ is fixed to be
nearly the Planck scale, the ratio between the relic and the present observed
vacuum energy density is $\rho_{vI}/\rho_{v0} \simeq 10^{123}$.
