  String dynamics in a curved space-time is studied on the basis of an action
functional including a small parameter of rescaled tension
$\epsilon=\gamma/\alpha^{\prime}$, where $\gamma$ is a metric parametrizing
constant. A rescaled slow worldsheet time $T=\epsilon\tau$ is introduced, and
general covariant non-linear string equation are derived.
  It is shown that in the first order of an $\epsilon $-expansion these
equations are reduced to the known equation for geodesic derivation but
complemented by a string oscillatory term. These equations are solved for the
de Sitter and Friedmann -Robertson-Walker spaces. The primary string
constraints are found to be split into a chain of perturbative constraints and
their conservation and consistency are proved. It is established that in the
proposed realization of the perturbative approach the string dynamics in the de
Sitter space is stable for a large Hubble constant $H
(\alpha^{\prime}H^{2}\gg1)$.
