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# Wydawnictwa / Czasopisma IMPAN / Applicationes Mathematicae / Wszystkie zeszyty

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

### Tom 41 / 2014

Applicationes Mathematicae 41 (2014), 107-129 MSC: 47J06, 47A52, 65J20, 65N20. DOI: 10.4064/am41-1-9

#### Streszczenie

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.

#### Autorzy

• 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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