A two step newton type iteration for ill-posed Hammerstein type operator equations in Hilbert scales

Abstract

In this paper regularized solutions of ill-posed Hammerstein type operator equation $KF(x)=y$, where $K:X\rightarrow Y$ is a bounded linear operator with non-closed range and $F:X\rightarrow X$ is non-linear, are obtained by a two step Newton type iterative method in Hilbert scales, where the available data is $y^{\delta}$ in place of actual data $y$ with $\|y-y^\delta\|\leq\delta$. We require only a weaker assumption $\|F^{\p}(x_0)x\|\sim\|x\|_{-b}$ compared to the usual assumption $\|F^{\p}(\widehat{x})x\|\sim\|x\|_{-b}$, where $\widehat{x}$ is the actual solution of the problem, which is assumed to exist, and $x_0$ is the initial approximation. Two cases, viz-a-viz, (i)~when $F^{\p}(x_0)$ is boundedly invertible and (ii)~$F^{\p}(x_0)$ is non-invertible but $F$ is monotone operator, are considered. We derive error bounds under certain general source conditions by choosing the regularization parameter by an a~priori manner as well as by using a modified form of the adaptive scheme proposed by Perverzev and Schock [{\bf14}].