This paper applies the asymptotic stability theory for ordinary differential equations to Gavurin's continuous analogue of several well-known nonlinear iterative methods. In particular, a general theory is developed which extends the Ortega-Rheinboldt concept of consistency to include the widely used finite difference approximations to the gradient as well as the finite difference approximation to the Jacobian in Newton's method. The theory is also shown to be applicable to the Levenberg-Marquardt methods.