Existence conditions of nonsingular dyonic black holes in nonlinear electrodynamics

General relativity coupled to nonlinear electrodynamics is known to have nonsingular black hole solutions. We investigate the existence conditions for such solutions in two-parameter Lagrangian ${\cal L} \left( {\cal F} , {\cal G} \right)$. In particular, we obtain a criterion on the Lagrangian for the existence of nonsingular black hole with a dyonic charge. In addition, we present a simple example of two-parameter Lagrangian satisfying the criterion, in which the existence of the dyonic solution is actually confirmed. Moreover, apart from the actual existence of dyonic solutions, we consider some examples for the Lagrangian satisfying such a criterion.