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 ? Y is a bounded linear operator with non-closed range and F : X ? X is non-linear, are obtained by a two step Newton type iterative method in Hilbert scales, where the available data is y? in place of actual data y with

y-y?

? ?. We require only a weaker assumption

F'(x<inf>0</inf>)x

?

x

<inf>-b</inf> compared to the usual assumption

F'(x?)x

?

x

<inf>-b</inf>, where x? is the actual solution of the problem, which is assumed to exist, and x<inf>0</inf> is the initial approximation. Two cases, viz-aviz, (i) when F'(x<inf>0</inf>) is boundedly invertible and (ii) F'(x<inf>0</inf>) 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 . © 2014 Rocky Mountain Mathematics Consortium.