Convergence Of Newton-steffensen Type Iteration Under Generalized Lipschitz Conditions And Its Application

This thesis studies the semilocal and local convergence of a Newton-Steffensen type iterative method under generalized Lipschitz conditions.The convergence results obtained in the thesis generalize the corresponding results in the relevant literatures.Specifically,they are described as follows:In chapter 1,an overview on the convergence of Newton's method and its modifications is given.In addition some preliminaries are also produced.In chapter 2,we study the semilocal convergence of the Newton-Steffensen type iteration method under generalized Lipschitz conditions by using the majorizing sequence technique.The convergence criterion and error estimate are obtained.Our convergence results allow us to obtain the results based on the L-average Lipschitz condition and so the classical Lipschitz condition ?-condition are obtained as special cases.Moreover,we also obtained a new convergence result under the H?lder conditions.In chapter 3,we use the majorizing sequence technique to study the local convergence of the Newton-Steffensen type iterative method.Our convergence results extend the corresponding results under the H?lder conditions in the related literatures.In the last chapter,a nonlinear boundary values problem of second order is considered to illustrate the application of our convergence results.