with density $\rho$ and pressure $p=w\rho$ is described by the Friedmann equations $$\label{gr-first}
H^2+\frac{\kappa}{a^2}=\frac{8\pi G\rho}3+\frac{\Lambda}3\, ,$$ $$\label{gr-second}
\frac{\ddot{a}}{a}=-\frac{4\pi G\rho}3(1+3w)+\frac{\Lambda}3\, ,$$ giving the following conditions of static solution: $$\frac{\kappa}{a_0^2}=\frac{8\pi G\rho_0}3+\frac{\Lambda}3\, ,$$ $$\frac{4\pi G\rho_0}3(1 + 3w) - \frac{\Lambda}3 = 0\, .$$ Introducing small homogeneous deviations from stationarity, $$a=a_0(1-\epsilon)\, , \quad \epsilon = \epsilon(t) \ll 1 \, ,$$ and linearizing the Friedmann equations (\[gr-first\]) and (\[gr-second\]) with respect to the small quantity $\epsilon
(t)$, we have: $$\frac{2\kappa\epsilon}{a_0^2} = \frac{8\pi G(\rho-\rho_0)}3\, ,$$ $$\label{gr}
\ddot{\epsilon} = W \epsilon, \quad W=\frac{\kappa(1+3w)}{a_0^2}\, .$$ We note that the weak energy condition and casuality requirements restrict the value of $w$ to lie in the interval $w \in [-1,+1]$ (see, e.g., [@HE]) and, as a result, $\kappa=+1$. Taking this into account, one can see from (\[gr\]) that the static solution in this theory is unstable for $w \in
\left[-\frac13,1\right]$, and stable for $w \in \left(-1,-\frac13\right)$.

The case of braneworld theory
-----------------------------

One can use the method outlined in the previous section to investigate the braneworld theory. In this section, we find (analytically and graphically) the regions of existence of static solutions in the parameter space and then further restrict the space of parameters by the requirement that the described theory 