The Convergence Behavior And Applications For Newton-Type Iterative Methods Under The Affine Transformation Conditions

This thesis is concerned with the semilocal convergence problems of some Newton-type iterative methods to solve nonlinear operator equations in Banach spaces. Under two classes of affine transformations conditions, some convergence results, which include and/or improve the existing relevant ones, are obtained.In Chapter 2, by using the so-called analysis approach of recurrence relations, we obtain two convergence results, which include Kantorovich-type convergence criterion, the R-order of convergence and the field of uniqueness of the solution, for Halley's method and a modified family of Halley-Chebyshev iteratives, respectively. For first one, the semilocal convergence properties of Halley's method for nonlinear operator equations are studied under the hypothesis that the second derivative satisfies some weak Lipschitz condition. And for the second one, the semilocal convergence of a family of Chebyshev-Halley like iterations for nonlinear operator equations is studied under the hypothesis that the first derivative satisfies a mild differentiability condition. The condition includes the usual Lipschitz condition and the Holder condition as special cases.In Chapter 3, under some majorant conditions which are weaker than the L-average Lipschitz conditions, a new semilocal convergence analysis for Halley's method is pre-sented. This analysis provides a clear relationship between the majorizing function and the nonlinear operator. This approach enables us to drop out the assumption of the existence of a second root for the majorizing function, but still guarantees Q-cubic con-vergence rate and obtain a new estimate of this rate based on a directional derivative of the twice derivative of the majorizing function. Moreover, the majorizing function does not have to be defined beyond its first root for obtaining convergence rate results.In Chapter 4, we study the semilocal convergence of Newton's method and simpli-fied Newton method under some weak affine Contravariant conditions. The results we obtained generalize the ones given by Deuflhard and Hohmann, respectively.And finally in Chapter 5, some applications to a nonlinear Hammerstein integral equation of the second kind are provided. And some numerical examples are presented to demonstrate the applicability and efficiency of the obtained convergence results.