Born - Infeld-like modified gravity
  A modified theory of gravity with the function $F(R) = (1-\sqrt{1-2\lambda
R})/\lambda$ is suggested and analyzed. At small value of the parameter
$\lambda$ introduced the action is converted into Einstein$-$Hilbert action.
The theory is consistent with local tests which gives a bound on the value of
the parameter $\lambda\leq 2\times 10^{-6}$ cm$^2$. I have considered the
Jordan frame as well as the Einstein frame in which the potential of the scalar
field was obtained. The static Schwarzschild$-$de Sitter solutions of the model
are obtained and analyzed. It was demonstrated that the de Sitter space is
unstable but a solution with zero Ricci scalar is stable. It is shown that
there is no matter instability in the proposed model.
