Convergence Analysis Of Inexact Newton-type Iteration Methods With Non-differentiable Nonlinear Equations

With the rapid development of technology and the popularization of computer ap-plication, the method for solving the nonlinear equation systems is widely used especially in engineering and economics. This thesis mainly concerns the semilocal and local con-vergence of inexact Newton-type methods solving non-differentiable nonlinear equation H(x)=0. Our work weakens some relevant convergence conditions and get some results. The contents are as follows:Chapter1, introduces the development of inexact Newton-type methods and rele-vant preliminary knowledge, such as basic concept, convergence of iteration, condition of convergence and relevant knowledge in Banach space. Using the majorizing sequence to prove semilocal convergence and two common methods to construct the majorizing sequence are also presented. The paper structure is shown at last.In Chapter2, Using inexact Newton-type methods to solve nonlinear operator equa-tion H(x)=0, when the derivative of H(x) doesn’t exist we divide H into differentiable part F(x) and non-differentiable part G(x). By using some majorizing sequences, its semilocal convergences are proved. The second Hammerstein integral equation is also prosented to illustrate the advantage of the obtained results.Chapter3, we study the local convergence of non-differentiable nonlinear equation H(x)=0. The differentiable part F(x) satisfies some weak Lipschitz conditions relaxed to a majorizing function, while the non-differentiable part G(x) satisfies Lipschitz condition. Then F(x) and G(x) satisfy Lipschitz condition as a special case is presented.