Bulk scalar field in brane-worlds with induced gravity inspired by the $\mathcal{L}(R)$ term

We obtain the effective field equations in a brane-world scenario within the framework of a DGP model where the action on the brane is an arbitrary function of the Ricci scalar, $\mathcal{L}(R)$, and the bulk action includes a scalar field in the matter Lagrangian. We obtain the Friedmann equations and acceleration conditions in the presence of the bulk scalar field for the $R^n$ term in four-dimensional gravity.