Regular multihorizon black holes in $f(G)$ gravity with nonlinear electrodynamics

In this work, we study the existence of regular black holes solutions with multihorizons in general relativity and in some alternative theories of gravity. We consider the coupling between the gravitational theory and nonlinear electrodynamics. The coupling generates modifications in the electromagnetic sector. This paper has as main objective generalize solutions already known from general relativity to the $f(G)$ theory. To do that, we first correct some misprints of the Odintsov and Nojiri's work in order to introduce the formalism that will be used in the $f(G)$ gravity. In order to satisfy all field equations, the method to find solutions in alternative theories generates different $f(R)$ and $f(G)$ functions for each solution, where only the nonlinear term of $f(G)$ contributes to the field equations. We also analyze the energy conditions, since it is expected that some must be violated to find regular black holes, and using an auxiliary field, we analyze the nonlinearity of the electromagnetic theory.