  We present nonsingular cosmological models with a variable cosmological term
described by the second-rank symmetric tensor $\Lambda_{mn}$ evolving from
$\Lambda g_{mn}$ to $\lambda g_{mn}$ with $\lambda < \Lambda$. All
$\Lambda_{mn}$ dominated cosmologies belong to Lemaitre type models for an
anisotropic perfect fluid. The expansion starts from a nonsingular
nonsimultaneous de Sitter bang, with $\Lambda$ on the scale responsible for the
earliest accelerated expansion, which is followed by an anisotropic Kasner type
stage. For a certain class of observers these models can be also identified as
Kantowski-Sachs models with regular R regions. For Kantowski-Sachs observers
the cosmological evolution starts from horizons with a highly anisotropic
``null bang'' where the volume of the spatial section vanishes. We study in
detail the spherically symmetric case and consider the general features of
cosmologies with planar and pseudospherical symmetries. Nonsingular
$\Lambda_{mn}$ dominated cosmologies are Bianchi type I in the planar case and
hyperbolic analogs of the Kantowski-Sachs models in the pseudospherical case.
At late times all models approach a de Sitter asymptotic with small $\lambda$.
