Always convergent iteration methods for nonlinear equations of Lipschitz functions

We define a class of always convergent methods for solving nonlinear equations of real Lipschitz functions. These methods generate monotone iterations that either converge to the nearest zero, if exists or leave the interval in a finite number of steps. We also investigate the speed and relative speed of these methods and show that no optimal method exists in the class. After deriving special cases we report on numerical testing that show the feasibility of these methods.