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We propose a large class of nonsingular cosmologies of arbitrary spatialcurvature whose cosmic history is determined by a primeval dynamical $\Lambda(t)$-term. For all values of the curvature, the models evolve between twoextreme de Sitter phases driven by the relic time-varying vacuum energydensity. The transition from inflation to the radiation phase is universal andpoints to a natural solution of the graceful exit problem regardless of thevalues of the curvature parameter. The flat case recovers the scenario recentlydiscussed in the literature (Perico et al., Phys. Rev. D88, 063531, 2013). Theearly 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 benearly the Planck scale, the ratio between the relic and the present observedvacuum energy density is $\rho_{vI}/\rho_{v0} \simeq 10^{123}$.

Decaying Vacuum Inflationary Cosmologies: A Complete Scenario Including Curvature Effects