Discretized Newton-Tikhonov method for ill-posed hammerstein type equations

Abstract

George and Shobha (2012) considered the finite dimensional realization of an iterative method for non-linear ill-posed Hammerstein type operator equation KF(x) = f, when the Fr chet derivative F' of the non-linear operator F is not invertible. In this pa- per we consider the special case i.e., F'-1 exists and is bounded. We analyze the convergence using Lipschitz-type conditions used in [10], [13], [22] and also analyze the convergence using a center type Lipschitz condition. The center type Lipschitz con- dition provides a tighter error estimate and expands the applicability of the method. Using a logarithmic-type source condition on F(x0)-F(?) (here ? is the actual solution of KF(x) = f) we obtain an optimal order convergence rate. Regularization param- eter is chosen according to the balancing principle of Pereverzev and Schock (2005). Numerical illustrations are given to prove the reliability of our approach.