enlarges the class of cosmological solutions, in particular, it allows for the existence of static cosmological solutions with ordinary matter content even in the case of spatially flat or open cosmology, a specific feature of the braneworld cosmology that was also noted in [@CS; @GM; @SS]. In the Randall–Sundrum model, flat or open static universe with ordinary matter requires the presence of the negative dark-radiation term, while, in the model with scalar-curvature term in the action for the brane to be discussed in this paper, the presence of dark radiation may be not necessary.

The paper is organized as follows:   In Sec. \[basic\], we introduce the basic equations describing the theory with one brane in the five-dimensional bulk space. Then, in Sec. \[static\], we introduce a method for the investigation of static solutions. Using this method, we describe both the general-relativity case and the case of braneworld theory that contains the scalar-curvature term in the action for the brane. In the partial case of spatially flat universe, we determine the range of parameters where static solutions exist and where they are stable with respect to homogeneous and isotropic perturbations. In Sec. \[nonstatic\], we develop an approach which is used to 