Globally convergent inexact generalized Newton's methods for nonsmooth equations

In this paper, motivated by the Martinez and Qi methods (J. Comput. Appl. Math. 60 (1995) 127), we propose one type of globally convergent inexact generalized Newton's methods to solve nonsmooth equations in which the functions are nondifferentiable, but are Lipschitz continuous. The methods make the norm of the functions decreasing. These methods are implementable and globally convergent. We also prove that the algorithms have superlinear convergence rates under some mild conditions.