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

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^δ in place of actual data y with ||y-y^δ|| ≤ δ. We require only a weaker assumption ||F'(x₀)x||̃ ||x||_-b compared to the usual assumption ||F'(x̂)x||̃ ||x||_-b, where x̂ is the actual solution of the problem, which is assumed to exist, and x₀ is the initial approximation. Two cases, viz-aviz, (i) when F^'(x₀) is boundedly invertible and (ii) F'(x₀) 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 .