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 sup(delta) in place of actual data y with ||y-y sup(delta)|| less than or equal to delta. We require only a weaker assumption ||F'(x sub(0))x||tilt||x||sub(−b) compared to the usual assumption ||F'(cap-x)x||tilt||x||sub(−b), where cap-x is the actual solution of the problem, which is assumed to exist, and x sub(0) is the initial approximation. Two cases, viz-a-viz, (i) when F'(x sub(0)) is boundedly invertible and (ii) F'(x sub(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.