Generalization of Regular Black Holes in General Relativity to $f(R)$ Gravity

In this paper, we determine regular black hole solutions using a very general $f(R)$ theory, coupled to a non-linear electromagnetic field given by a Lagrangian ${\mathcal{L}}_{NED}$. The functions $f(R)$ and ${\mathcal{L}}_{NED}$ are left in principle unspecified. Instead, the model is constructed through a choice of the mass function $M(r)$ presented in the metric coefficients. Solutions which have a regular behaviour of the geometric invariants are found. These solutions have two horizons, the event horizon and the Cauchy horizon. All energy conditions are satisfied in the whole space-time, except the strong energy condition (SEC) which is violated near the Cauchy horizon.