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Motivated by the two-step directional Newton method considered by Argyros and Hilout (2010) for approximating a zero    of a differentiable function  F  defined on a convex set    of a Hilbert space  H , we consider a two-step Newton–Lavrentiev method (TSNLM) for obtaining an approximate solution to the nonlinear ill-posed operator equation   , where    is a nonlinear monotone operator defined on a real Hilbert space  X . It is assumed that    and that the only available data are    with   . We prove that the TSNLM converges cubically to a solution of the equation    (such solution is an approximation of   ) where  x 0   is the initial guess. Under a general source condition on   , we derive order optimal error bounds by choosing the regularization parameter α according to the balancing principle considered by Perverzev and Schock (2005). The computational results provided endorse the reliability and effectiveness of our method.

An application of Newton-type iterative method for the approximate implementation of Lavrentiev regularization