On Newton-like methods for solving nonlinear equations

In this paper, we present a family of general Newton-like methods with a parametric function for finding a zero of a univariate function, permittingf′(x)=0 in some points. The case of multiple roots is not treated. The methods are proved to be quadratically convergent provided the weak condition. Thus the methods remove the severe conditionf′ (x)≠0. Based on the general form of the Newton-like methods, a family of new iterative methods with a variable parameter are developed.