  In this paper we investigate cosmological dynamics on the normal branch of a
DGP-inspired scenario within a phase space approach where induced gravity is
modified in the spirit of $f(R)$-theories. We apply the dynamical system
analysis to achieve the stable solutions of the scenario in the normal DGP
branch. Firstly, we consider a general form of the modified induced gravity and
we show that there is a standard de Sitter point in phase space of the model.
Then we prove that this point is stable attractor only for those $f(R)$
functions that account for late-time cosmic speed-up.
