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

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