On the method of Lavrentiev regularization and majorizing sequence for solving ill-posed problems with monotone operators

Abstract

In this paper we consider the Lavrentiev regularization method for obtaining stable approximate solution to nonlinear ill-posed operator equations F (x) = y where F : D(F ) X ! X is a nonlinear monotone operator dened on a real Hilbert space X. We assume that only a noisy data y 2 X withky y k are available. Under the assumption that F is Lipschitz continuous, the iteration x n; converges to the unique solution x of the equation F (x) + (x x0) = y (x0 := x 0; ). It is known that (Tautanhahn (2002)) x converges to the solution ^ of F (x) = y: The convergence analysis and the stopping rule are based on a suitably constructed majorizing sequence. Under a general source condition on x0 ^ x we proved that the errorkx n; ^