The Convergence Analysis Of Some Iterative Methods For Solving Nonlinear Equations In Banach Spaces

Solving nonlinear equations or nonlinear systems of equations F(x)= 0 is a class of important problems in modern mathematics research,which has the practical significance and scientific value to solve real problem in Hi-Tech times and future.A common method to solve this kind of ques-tions is iterative method.However,the results of a nonlinear problem will be directly affected by the choice of the iterative method.Thus it is very important and necessary to study iterative methods.In this thesis,we investigate the convergence of two Secant-like itera-tions and a deformed Newton iteration.The whole article consists of four chapters.In the first chapter,by analyzing and summarizing the achieve-ments of domestic and foreign researchers in this domain,the article ex-pounds the significance and practical background of solving nonlinear equa-tions by iterative methods.At the same time,we present the historical developments in convergence analysis for Secant-like methods and the de-formed Newton method.In the second chapter,we study the convergence of two points Secant method and single point Secant method under the condition of the radius Lipschitz condition with the L average.The weak Smale's point estimate condition and Kantorovich-type condition are unified by this condition. The proofs of the existence and convergence theorems are given.In the third chapter,we establish a local convergence theorem of the two points Secant method through the use of the recurrence technique.By improving the Holder condition which the divided differences of order one satisfy,we obtain radius of convergence ball the uniqueness ball.All the theoretical conclusions are tested and verified by a numerical example.In the fourth chapter,we derive two majorizing functions and discuss the convergence of deformed Newton iteration for solving the nonlinear equations with non-differentiable terms H(x)+G(x)=0 by using the ma-jorizing sequences.We also present the local and semilocal convergence theorems for the deformed Newton method.