the braneworld theory. First, we review the situation in the general relativity theory, where the static Einstein’s universe is known to be unstable. We apply the same method for the investigation of the stability of static solutions in the braneworld theory and demonstrate new possibilities arising in this theory. Firstly, unlike general relativity, the braneworld theory admits static solutions even in the case of spatially flat or open universe, which, in the context of the Randall–Sundrum model, was also noted in [@GM]. In the Randall–Sundrum model, static spatially flat or open universes with matter satisfying the strong energy condition are possible only with negative value of the dark-radiation constant $C$, while, in the presence of the scalar-curvature term in the action for the brane, a spatially flat or open universe can be static even when this constant is zero. Secondly, these solutions can be stable in certain region of parameters.

The case of general relativity
------------------------------

In this section, we review the situation in the general relativity theory with the standard action $$S_{\rm GR}=-\frac{1}{16\pi G}\int(R-2\Lambda)\sqrt{-g}d^4x+\int L_{\rm
matter}\sqrt{-g}d^4x$$ and describe our method for investigation of the stability of static solutions on the example of this theory. A homogenous and isotropic universe 