  We investigate the Robertson-Walker cosmology with Lagrangian
$R+\alpha_1\hbar R^2+\alpha_2\hbar R^{\mu\nu}R_{\mu\nu} +L_{rad}$ where
$L_{rad}$ means classical source with traceless energy-momentum tensor. We
weaken the self-consistence condition (L. Parker, J. Z. Simon, Phys. Rev.
{D47}(1993),{1339}). Quantum corrections are expressed as contributions to the
effective equation of state. We show that the empty space-time is stable within
the class of radiation-filled expanding universes with no order reduction of
the field equations.
