  The stability criteria for the generalized Brans-Dicke cosmology in a
spatially flat, homogeneous and isotropic cosmological model is discussed in
the presence of a perfect fluid. The generalization comes through the channel
that the Brans-Dicke coupling parameter $\omega$ is allowed to be a function of
the scalar field $\phi$. This generalization can lead to a host of
scalar-tensor theories of gravity for various choices of $\omega = \omega
(\phi)$. A very interesting general result has been found. Excepting for the
case of a radiation distribution as the choice of the fluid, all other
solutions find a natural habitat in the corresponding solutions in general
relativity in an infinite $\omega$ limit. For the radiation distribution, the
dependence of stability on $\omega$ is a bit obscure. If a scalar potential,
function of the Brans-Dicke scalar field, is added to the action, the
requirement of an infinite $\omega$ for stability is relaxed for a matter
distribution with a non-zero trace whereas it becomes a possibility for a
radiation distribution.
