Weaker convergence conditions of an iterative method for nonlinear ill-posed equations

Abstract

In this chapter we expand the applicability of an iterative method which converges to the unique solution xα of the method of Lavrentiev regularization, i.e., F(x) + α(x - x0) = y, approximating the solution x of the ill-posed problem F(x) = y where F: D(F) - X - X is a nonlinear monotone operator defined on a real Hilbert space X. We use a center-Lipschitz instead of a Lipschitz condition used in [1-3]. The convergence analysis and the stopping rule are based on the majorizing sequence. The choice of the regularization parameter is the crucial issue. We show that the adaptive scheme considered by Perverzev and Schock [4] for choosing the regularization parameter can be effectively used here for obtaining order optimal error estimate. Numerical examples are presented to show that older convergence conditions [1-3] are not satisfied but the new ones are satisfied. © 2020 by Nova Science Publishers, Inc. All rights reserved.