used in the analysis by Campos and Sopuerta [@CS] of the Randall–Sundrum braneworld theory. In this section, we extend their investigation to the case where the scalar-curvature term is also present in the action for the brane, but we restrict ourselves to the homogeneous and isotropic cosmology.

 The case of the Randall–Sundrum braneworld model {#method}
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In this subsection, we briefly describe the method and results of [@CS], where the model with $m=0$ was under consideration. After setting $m = 0$, Eqs. (\[second\]) and (\[first\]) turn to $$\label{m2}
9M^{6}\left[\frac{\ddot{a}}{a}+H^2+\frac{\kappa}{a^2}-\frac{\Lambda_{\rm
b}}3\right] +[\rho(1+3w)-2\sigma](\rho+\sigma)=0$$ and $$\label{m1}
H^2+\frac{\kappa}{a^2} = \frac{(\rho+\sigma)^2}{9M^{6}} + \frac{C}{a^4} +
\frac{\Lambda_{\rm b}}{6}\, ,$$ respectively. Here, for simplicity, we consider the case $C > 0$. The last equation can be written as follows: $$H^2 = \frac{2\sigma}{9M^{6}} \rho \left(1 + \frac{\rho}{2\sigma} \right) -
\frac{\kappa}{a^2} + \frac{C}{a^4} + \frac13 \left( \frac{\Lambda_{\rm
b}}2+\frac{\sigma^2}{3M^{6}}\right) \, .$$

Using the definition of the four-dimensional cosmological constant $$\Lambda_{\rm RS} = \frac{\Lambda_{\rm b}}2+\frac{\sigma^2}{3M^{6}}\, ,$$ which is proportional to the Randall–Sundrum constraint [@RS], we introduce the notation similar to those of [@CS]: $$\Omega_{\rho}=\frac{2\sigma\rho}{9M^{6}H^2}\, , \quad
\Omega_{k}=-\frac{\kappa}{a^2H^2}\, , \quad \Omega_{\Lambda}=\frac{\Lambda_{\rm
RS}}{3H^2}\, , \quad \Omega_{\lambda}=\frac{\rho^2}{9M^{6}H^2}\, , \quad
\Omega_C = \frac{C}{a^4 H^2} \, .$$

The authors of [@CS] work in the five-dimensional $\Omega$-space $(\Omega_{\rho},\Omega_{k},\Omega_{\Lambda},\Omega_{\lambda},\Omega_C)$. 