  We investigate the modified $F(R)$ gravity theory with the function $F(R) =
(1-\sqrt{1-2\lambda R-\sigma (\lambda R)^2})/\lambda$. The action is converted
into Einstein$-$Hilbert action at small values of $\lambda$ and $\sigma$. The
local tests give a bound on the parameters, $\lambda(1+\sigma)\leq 2\times
10^{-6}$ cm$^2$. The Jordan and Einstein frames are considered, the potential,
and the mass of the scalar field were obtained. The constant curvature
solutions of the model are found. It was demonstrated that the de Sitter space
is unstable but a solution with zero Ricci scalar is stable. The cosmological
parameters of the model are evaluated. Critical points of autonomous equations
are obtained and described.
