Newton Lavrentiev regularization for ill-posed operator equations in Hilbert scales

In this paper we consider the two step method for approximately solving the ill-posed operator equation F(x)=f,F(x)=f, where F:D(F)⊆X→X,F:D(F)⊆X→X, is a nonlinear monotone operator defined on a real Hilbert space X,X, in the setting of Hilbert scales. We derive the error estimates by selecting the regularization parameter αα according to the adaptive method considered by Pereverzev and Schock in (2005), when the available data is fδfδ with ‖f-fδ‖⩽δ‖f-fδ‖⩽δ. The error estimate obtained in the setting of Hilbert scales {Xr}r∈R{Xr}r∈R generated by a densely defined, linear, unbounded, strictly positive self adjoint operator L:D(L)⊂X→XL:D(L)⊂X→X is of optimal order.