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Newton-type iterative methods for nonlinear ill-posed Hammerstein-type equations

Volume 41 / 2014

Monnanda Erappa Shobha, Ioannis K. Argyros, Santhosh George Applicationes Mathematicae 41 (2014), 107-129 MSC: 47J06, 47A52, 65J20, 65N20. DOI: 10.4064/am41-1-9

Abstract

We use a combination of modified Newton method and Tikhonov regularization to obtain a stable approximate solution for nonlinear ill-posed Hammerstein-type operator equations $KF(x)=y.$ It is assumed that the available data is $y^\delta $ with $\| y-y^\delta \| \leq \delta ,$ $ K:Z\rightarrow Y$ is a bounded linear operator and $ F:X\rightarrow Z $ is a nonlinear operator where $X,Y,Z$ are Hilbert spaces. Two cases of $F$ are considered: where $F'(x_0)^{-1}$ exists ($F'(x_0)$ is the Fréchet derivative of $F$ at an initial guess $x_0$) and where $F$ is a monotone operator. The parameter choice using an a priori and an adaptive choice under a general source condition are of optimal order. The computational results provided confirm the reliability and effectiveness of our method.

Authors

  • Monnanda Erappa ShobhaDepartment of Mathematical
    and Computational Sciences
    National Institute of Technology
    Karnataka, India 757 025
    e-mail
  • Ioannis K. ArgyrosDepartment of Mathematical Sciences
    Cameron University
    Lawton, OK 73505, U.S.A.
    e-mail
  • Santhosh GeorgeDepartment of Mathematical and Computational Sciences
    National Institute of Technology
    Karnataka, India 757 025
    e-mail

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