Newton type iteration for Tikhonov regularization of non-linear ill-posed Hammerstein type equations

An iterative method is investigated for a nonlinear ill-posed Hammerstein type operator equation KF(x)=f. We use a center-type Lipschitz condition in our convergence analysis instead of the usual Lipschitz condition. The adaptive method of Pereverzev and Schock (SIAM J. Numer. Anal. 43(5):2060–2076, 2005) is used for choosing the regularization parameter. The optimality of this method is proved under a general source condition involving the Fréchet derivative of F at some initial guess x ₀. A numerical example of nonlinear integral equation shows the efficiency of this procedure.