Iterative Methods and their Applications for Solving Non-Linear Ill-Posed Equations

Abstract

This thesis deals with iterative methods and their convergence for solving non-linear equations in Banach Spaces. As an application, it also deals with solving non-linear ill posed equations in a Hilbert space setting. Under various assumptions, local and semi local convergence analyses of some iterative schemes are studied. We have established the desired order of convergence using weaker assumptions than those available in the literature. We have also extended some of the methods efficiently. Computable radii of convergence and dynamics analysis using the basin of attractions are other highlights. The first contribution of the thesis is the convergence analysis of a fifth-order it erative method using conditions only on the first Fréchet derivative. This increased the applicability of the method. In our second work, we used the iterative method for solving the regularized equation corresponding to a non-linear ill-posed equation. We introduced a new source condition and parameter choice strategy for the desired results. Thirdly, using Lipschitz-type assumptions on first and second derivatives instead of Taylor series expansion, we established third-order convergence of an iterative Home ier method. We further extended this method to the fifth and sixth order. Lastly, we studied another iterative method introduced by Traub. We established third-order con vergence without using Taylor series expansion. We extended this method to the fifth and sixth order.