The Pure and Applied Mathematics (한국수학교육학회지시리즈B:순수및응용수학)

Korean Society of Mathematical Education (한국수학교육학회)
권호 표지

ON THE CONVERGENCE OF NEWTON'S METHOD AND LOCALLY HOLDERIAN INVERSES OF OPERATORS

  • Published : 2009.02.28
Download PDF
⟨ Previous Next ⟩

Abstract

A semilocal convergence analysis is provided for Newton's method in a Banach space. The inverses of the operators involved are only locally $H{\ddot{o}}lderian$. We make use of a point-based approximation and center-$H{\ddot{o}}lderian$ hypotheses for the inverses of the operators involved. Such an approach can be used to approximate solutions of equations involving nonsmooth operators.

Keywords

References

  1. Argyros, I.K. : Advances in the Effciency of Computational Methods and Applications. World Scientific Publ. Co., River Edge, NJ, 2000.
  2. Argyros, I.K. : An improved error analysis for Newton-like methods under generalized conditions. J. Comput. Appl. Math. 157 (2003), no. 1, 169-185. https://doi.org/10.1016/S0377-0427(03)00390-X
  3. Argyros, I.K. : On the convergence of Newton's method and locally Holderian operators. J. Korea Soc. Math. Edu. Ser. B: Pure Appl. Math. 15 (2008), no. 2, 111-120.
  4. Kantorovich, L.V. & Akilov, G.P.: Functional Analysis in Normed Spaces. Pergamon Press, Oxford, 1982.
  5. Robinson, S.M.: An implicit function theorem for a class of nonsmooth functions. Mathematics of Operations Research, 16 (1991), no. 2, 292-309. https://doi.org/10.1287/moor.16.2.292
  6. Robinson, S.M.: Newton's method for a class of nonsmooth functions. Set- Valued Analysis, 2 (1994), 291-305. https://doi.org/10.1007/BF01027107