The Convergence Analysis And Application Of Several Iterative Methods For Solving Nonlinear Equations In Banach Space

The study of nonlinear problems is an active field in modern mathematical research. Many nonlinear problems, including nonlinear mathematical physics, nonlinear finite element, non-linear mechanics, the power system, economic and nonlinear programming, can be reduced to solving nonlinear equations in Banach space. Iterative method presents the most important and convenient way to obtain approximate solutions of nonlinear equations, the convergence anal-ysis of which remains fundamental but crucial. In this sense, the research on the convergence analysis of the iterative methods is of importance in both theory and practice.This thesis consists of five chapters, with emphasis on the numerical performance, the con-vergence rate, error analysis and applications in nonlinear equations of some iterative methods. The main work are displayed below:In Chapter1, with the background of solving nonlinear equations in place, several popular iterative methods are introduced. Some analysis of previous work, as well as convergence conditions and techniques for validating them, are also given. For ease of clarifying subsequent chapters, theoretical results including some theorems and lemmas are also provided.In Chapter2, the semilcoal convergence of super-Halley method via majoring functions is investigated. The convergence theorem and error estimate show that the proposed method is superior to that obtained through recurrence relations in that it has lager convergence ra-dius. Furthermore, the semilocal convergence of super-Halley method in weak conditions is also studied.In Chapter3, under Lipschitz continuous conditions or Holder continuous conditions, the semilocal convergence of some fifth-order methods are examined in terms of convergence order and the error estimate. Further, a family of fifth-order iterative methods with parameter which exploits the first derivative of divided differences instead of second derivative is developed. Its semilocal convergence under ω continuous conditions is studied through recurrence relations. Finally, the effectiveness of the proposed method is verified in solving nonlinear integral equa-tions.In Chapter4, a sixth-order method in real space is presented. Further, the method is extended to Banach space, and its semilocal convergence is established by recurrence relations. An existence-uniqueness theorem is given to demonstrate the R-order of the method to be six and a priori error bounds is also obtained. Finally, numerical results show this method is efficient.In Chapter5, a fifth-order method in real space is contrived, which is subsequently modified to a new method with parameter of order six. Then this family of sixth-order methods is then generalized to Banach space where the semilocal convergence under ω continuous conditions is checked by recurrence relations. The effectiveness of the new method is illustrated by applying it to solve mix Hammerstein-type nonlinear integral equations.