Black holes of multiple horizons without mass inflation

Mass inflation is a phenomenon happened in the vicinity of inner horizon in two-horizon spacetime. It is shown that the mass of initially small perturbations will grow exponentially as they approach the inner horizon. This implies that black hole inner horizon is unstable to the perturbations. In view of the fact that the mass inflation is determined by the surface gravity of inner horizon, Carballo-Rubio et al. showed that if one makes the surface gravity of inner horizon vanish, then the exponential growth character of mass inflation is not present. Basing on this conclusion, we look for the black hole solutions of multiple horizons with nonlinear Maxwell field. Then we make the inner horizons to coincide with each other such that the surface gravities of every inner horizon vanish. By this way, black holes of multiple horizons without mass inflation are constructed.