Born - Infeld-type modified gravity

We propose and investigate the modified Born$-$Infeld-type gravity model with the function $F(R) = [1-(1-\beta R/\sigma)^\sigma]/\beta$. At different values of the dimensionless parameter $\sigma$ the action is converted into some models including general relativity ($\sigma=1$), the Starobinsky model ($\sigma=2$), the exponential model of gravity ($\sigma=\infty$), and a model proposed in [9], [10] ($\sigma=0.5$). A bound on the parameters, $\beta(1-\sigma)/\sigma\leq 2\times 10^{-6}$ cm$^2$, are due to local tests. The constant curvature solutions are obtained, and we found the potential, and the mass of the scalaron in the Einstein frame. The model describes the inflation of the universe. The de Sitter spacetime is unstable and a Minkovsky spacetime is stable. We investigate the cosmological parameters and some critical points of autonomous equations.