Convergence Analysis Of Two-step Iterative Method For Non-differentiable Operators

In physics and engineering,it is frequent to use iterative methods for finding approximate solutions to the established mathematical models.Firstly,one discretizes a mathematical model and gets a nonlinear operator equation;then starts from one or more initial values to obtain a convergent iterative sequence;finally chooses some in the sequence as approximate solutions.The nonlinear operator this thesis involved takes the form H?x?=F?x?+G?x?,where F?x?is a differentiable operator and G?x?is a non-differentiable operator.We mainly study the semi-local and local convergence of a two-step iterative method for solving H?x?=0.The layout is as follows:The first chapter introduces the development process of two-step iterative methods,and gives the basic concepts and related knowledge applied.The main results of this thesis are given at the last part of this chapter.The second chapter discusses the semilocal convergence of a two-step iterative method.In details,when the first derivative of the nonlinear operator F and the first-order difference quotient of the nonlinear operator G satisfy some ? conditions,the semilocal convergence theorem and the uniqueness theorem are proved,which generalizes some known results when the first derivative of F and the first-order difference quotient of G satisfies Holder condition.Two numerical examples are used to illustrate the rationality of the method.The third chapter studies the local convergence theorem and the uniqueness theorem of the two-step iterative method when the first derivative of F and the first-order difference quotient of G satisfy some ? conditions.The effectiveness of this method is verified by a numerical example.At the same time,by using the proof method of our theorem,a local convergence result of a King-Werner two-step iterative method with a convergence order of 1+2^1/2 is obtained under these ? conditions.