Before revision:  More specifically, we first propose an adaptive formula for the Levenberg-Marquardt parameter and analyse the local convergence of the method under H\"{o}lder metric subregularity . We then introduce a bounded version of the Levenberg-Marquardt parameter and analyse the local convergence of the modified method under the \L ojasiewicz gradient inequality  .
After revision:  More specifically, we first propose an adaptive formula for the Levenberg-Marquardt parameter and analyse the local convergence of the method under H\"{o}lder metric subregularity of the function defining the equation and H\"older continuity of its gradient. Further, we analyse the local convergence of the modified method under the \L ojasiewicz gradient inequality .