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 d
riven 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 $rho_{vI}/rho_{v0} simeq 10^{123}$.
