Table of Contents
ISRN Applied Mathematics
Volume 2012 (2012), Article ID 783579, 22 pages
Research Article

Two-Step Newton-Tikhonov Method for Hammerstein-Type Equations: Finite-Dimensional Realization

Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka, Mangalore 575 025, India

Received 26 December 2011; Accepted 26 January 2012

Academic Editors: A. Bellouquid, A. El-Sayed, A. J. Kearsley, J. MΓ­guez, and J. Shen

Copyright Β© 2012 Santhosh George and Monnanda Erappa Shobha. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Finite-dimensional realization of a Two-Step Newton-Tikhonov method is considered for obtaining a stable approximate solution to nonlinear ill-posed Hammerstein-type operator equations 𝐾𝐹(π‘₯)=𝑓. Here 𝐹∢𝐷(𝐹)βŠ†π‘‹β†’π‘‹ is nonlinear monotone operator, πΎβˆΆπ‘‹β†’π‘Œ is a bounded linear operator, 𝑋 is a real Hilbert space, and π‘Œ is a Hilbert space. The error analysis for this method is done under two general source conditions, the first one involves the operator 𝐾 and the second one involves the FrΓ©chet derivative of 𝐹 at an initial approximation π‘₯0 of the the solution Μ‚π‘₯: balancing principle of Pereverzev and Schock (2005) is employed in choosing the regularization parameter and order optimal error bounds are established. Numerical illustration is given to confirm the reliability of our approach.