Improved convergence conditions of a
Lavrentiev-type method for nonlinear ill-posed equations by using restricted convergence domains

Abstract. In this paper we extend the applicability of an iterative method which converges to the unique solution \(x_\alpha^\delta\) of the Lavrentiev regularization equation, i.e., \(F(x)+\alpha(x-x_0) = y^\delta,\) approximating the solution \(\hat{x}\) of the ill-posed problem \(F(x)= y\) where \(F:D(F)\subseteq X\longrightarrow X\) is a nonlinear monotone operator defined on a real Hilbert space \(X\). We use a center-Lipschitz instead of a Lipschitz condition used in [8, 15, 22] as well as our new idea of restricted convergence domains. This idea helps us determine a smaller ball where the iterates lie leading to smaller Lipschitz constants which in turn helps us provide a wider convergence domain, tighter error bounds on the distances involved and an at least as precise information on the location of the solution. These advantages are obtained under the same computational cost, since in practice the computation the old constants requires the computation of the new constants as special cases. 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 [19] for choosing the regularization parameter can be effectively used here for obtaining order optimal error estimate.