  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.
