`Abstract and Applied AnalysisVolume 2012 (2012), Article ID 982925, 13 pageshttp://dx.doi.org/10.1155/2012/982925`
Research Article

## Semilocal Convergence Analysis for Inexact Newton Method under Weak Condition

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

Received 29 May 2012; Accepted 5 August 2012

Copyright © 2012 Xiubin Xu et al. 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.

1. I. K. Argyros, â€śOn the Newton-Kantorovich hypothesis for solving equations,â€ť Journal of Computational and Applied Mathematics, vol. 169, no. 2, pp. 315â€“332, 2004.
2. J. A. Ezquerro and M. A. Hernández, â€śGeneralized differentiability conditions for Newton's method,â€ť IMA Journal of Numerical Analysis, vol. 22, no. 2, pp. 187â€“205, 2002.
3. J. A. Ezquerro and M. A. Hernández, â€śOn an application of Newton's method to nonlinear operators with $w$-conditioned second derivative,â€ť BIT. Numerical Mathematics, vol. 42, no. 3, pp. 519â€“530, 2002.
4. J. M. Gutiérrez, â€śA new semilocal convergence theorem for Newton's method,â€ť Journal of Computational and Applied Mathematics, vol. 79, no. 1, pp. 131â€“145, 1997.
5. J. M. Gutiérrez and M. A. Hernández, â€śNewton's method under weak Kantorovich conditions,â€ť IMA Journal of Numerical Analysis, vol. 20, no. 4, pp. 521â€“532, 2000.
6. M. A. Hernández, â€śThe Newton method for operators with Hölder continuous first derivative,â€ť Journal of Optimization Theory and Applications, vol. 109, no. 3, pp. 631â€“648, 2001.
7. X. H. Wang, â€śConvergence of Newton's method and inverse function theorem in Banach space,â€ť Mathematics of Computation, vol. 68, no. 225, pp. 169â€“186, 1999.
8. X. H. Wang, â€śConvergence of Newton's method and uniqueness of the solution of equations in Banach space,â€ť IMA Journal of Numerical Analysis, vol. 20, no. 1, pp. 123â€“134, 2000.
9. X. H. Wang and C. Li, â€śConvergence of Newton's method and uniqueness of the solution of equations in Banach spaces. II,â€ť Acta Mathematica Sinica, English Series, vol. 19, no. 2, pp. 405â€“412, 2003.
10. R. S. Dembo, S. C. Eisenstat, and T. Steihaug, â€śInexact Newton methods,â€ť SIAM Journal on Numerical Analysis, vol. 19, no. 2, pp. 400â€“408, 1982.
11. I. K. Argyros, â€śA new convergence theorem for the inexact Newton methods based on assumptions involving the second Fréchet derivative,â€ť Computers & Mathematics with Applications, vol. 37, no. 7, pp. 109â€“115, 1999.
12. Z. Z. Bai and P. L. Tong, â€śAffine invariant convergence of the inexact Newton method and Broyden's method,â€ť Journal of University of Electronic Science and Technology of China, vol. 23, no. 5, pp. 535â€“540, 1994.
13. J. Chen and W. Li, â€śConvergence behaviour of inexact Newton methods under weak Lipschitz condition,â€ť Journal of Computational and Applied Mathematics, vol. 191, no. 1, pp. 143â€“164, 2006.
14. M. G. Gasparo and G. Morini, â€śInexact methods: forcing terms and conditioning,â€ť Journal of Optimization Theory and Applications, vol. 107, no. 3, pp. 573â€“589, 2000.
15. X. P. Guo, â€śOn semilocal convergence of inexact Newton methods,â€ť Journal of Computational Mathematics, vol. 25, no. 2, pp. 231â€“242, 2007.
16. C. Li and W. P. Shen, â€śLocal convergence of inexact methods under the Hölder condition,â€ť Journal of Computational and Applied Mathematics, vol. 222, no. 2, pp. 544â€“560, 2008.
17. I. Moret, â€śA Kantorovich-type theorem for inexact Newton methods,â€ť Numerical Functional Analysis and Optimization, vol. 10, no. 3-4, pp. 351â€“365, 1989.
18. J. M. Martínez and L. Q. Qi, â€śInexact Newton methods for solving nonsmooth equations,â€ť Journal of Computational and Applied Mathematics, vol. 60, no. 1-2, pp. 127â€“145, 1995.
19. B. Morini, â€śConvergence behaviour of inexact Newton methods,â€ť Mathematics of Computation, vol. 68, no. 228, pp. 1605â€“1613, 1999.
20. W. P. Shen and C. Li, â€śConvergence criterion of inexact methods for operators with Hölder continuous derivatives,â€ť Taiwanese Journal of Mathematics, vol. 12, no. 7, pp. 1865â€“1882, 2008.
21. W. P. Shen and C. Li, â€śKantorovich-type convergence criterion for inexact Newton methods,â€ť Applied Numerical Mathematics, vol. 59, no. 7, pp. 1599â€“1611, 2009.
22. W. P. Shen and C. Li, â€śSmale's $\alpha$-theory for inexact Newton methods under the $\gamma$-condition,â€ť Journal of Mathematical Analysis and Applications, vol. 369, no. 1, pp. 29â€“42, 2010.
23. M. Wu, â€śA new semi-local convergence theorem for the inexact Newton methods,â€ť Applied Mathematics and Computation, vol. 200, no. 1, pp. 80â€“86, 2008.
24. T. J. Ypma, â€śLocal convergence of inexact Newton methods,â€ť SIAM Journal on Numerical Analysis, vol. 21, no. 3, pp. 583â€“590, 1984.