The Inexact Newton-like Methods For Nonlinear Equations

The solving problem of nonlinear operator equation f(x)= 0 is not only a very important mathematical problem in computational mathematics,but also has a wide range of practical applications in physics,economics,engineering,the field of life science and so on.And the iterative algorithm is an important tool to solve the problem.This paper mainly concerns the semilocal convergence problem of inexact Newton-like methods for solving nonlinear operator equation f(x)= 0.The contents are as follows:In Chapter 1,we introduce the development of iterations for solving nonlinear op-erator equations and the relevant preliminary knowledge,including the iterative form of inexact Newton-like methods,convergence condition,order of convergence,and related conclusions in Banach space.Using the majorizing sequence to prove semilocal conver-gence is also presented.The paper structure is shown at last.In Chapter 2,we study solving nonlinear operator equation f(x)= 0 by inexact Newton-like methods,if nonlinear operator f satisfies ?-condition,a semilocal conver-gence criterion for inexact Newton-like methods is established.And by using majorizing sequence,the semilocal convergence is proved.Finally,a numerical example is presented to illustrate the effectiveness of our results.In Chapter 3,under the weakened Kantorovich-type convergence criterion,we intro-duce center-Lipschitz condition and Lipschitz condition,and the semilocal convergence of inexact Newton-like methods is obtained.The main theorem obtained in this section expands the range of convergence and improves the related results.