A TWO STEP NEWTON TYPE ITERATION FOR ILL-POSED HAMMERSTEIN TYPE OPERATOR EQUATIONS IN HILBERT SCALES

摘要

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-≤ δ. We require only a weaker assumption ||F'(xo)x|| ?||x||-b compared to the usual assumption ||F'(x)x||?||x||_(-b), where 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'(x0) is boundedly invertible and (ii) F'(x0) is non-invert ible 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 [14].