Local convergence analysis of inexact Newton-like methods under majorant   condition
  We present a local convergence analysis of inexact Newton-like methods for solving nonlinear equations under majorant conditions. This analysis provides an estimate of the convergence radius and a clear relationship between the majorant function, which relaxes the Lipschitz continuity of the derivative, and the nonlinear operator under consideration. It also allow us to obtain some important special cases 
O. P. Ferreira and M. L. N. Goncalves