Open Access

Iterative Algorithm for Approximating Solutions of Maximal Monotone Operators in Hilbert Spaces

Fixed Point Theory and Applications20072007:032870

DOI: 10.1155/2007/32870

Received: 11 October 2006

Accepted: 11 December 2006

Published: 7 February 2007

Abstract

We first introduce and analyze an algorithm of approximating solutions of maximal monotone operators in Hilbert spaces. Using this result, we consider the convex minimization problem of finding a minimizer of a proper lower-semicontinuous convex function and the variational problem of finding a solution of a variational inequality.

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Authors’ Affiliations

(1)
Department of Mathematics, Tianjin Polytechnic University

References

  1. Martinet B: Régularisation d'inéquations variationnelles par approximations successives. Revue Française d'Automatique et Informatique. Recherche Opérationnelle 1970, 4: 154–158.MATHMathSciNetGoogle Scholar
  2. Rockafellar RT: Monotone operators and the proximal point algorithm. SIAM Journal on Control and Optimization 1976,14(5):877–898. 10.1137/0314056MATHMathSciNetView ArticleGoogle Scholar
  3. Brézis H, Lions P-L: Produits infinis de résolvantes. Israel Journal of Mathematics 1978,29(4):329–345. 10.1007/BF02761171MATHMathSciNetView ArticleGoogle Scholar
  4. Güler O: On the convergence of the proximal point algorithm for convex minimization. SIAM Journal on Control and Optimization 1991,29(2):403–419. 10.1137/0329022MATHMathSciNetView ArticleGoogle Scholar
  5. Passty GB: Ergodic convergence to a zero of the sum of monotone operators in Hilbert space. Journal of Mathematical Analysis and Applications 1979,72(2):383–390. 10.1016/0022-247X(79)90234-8MATHMathSciNetView ArticleGoogle Scholar
  6. Solodov MV, Svaiter BF: Forcing strong convergence of proximal point iterations in a Hilbert space. Mathematical Programming 2000,87(1):189–202.MATHMathSciNetGoogle Scholar
  7. Kamimura S, Takahashi W: Approximating solutions of maximal monotone operators in Hilbert spaces. Journal of Approximation Theory 2000,106(2):226–240. 10.1006/jath.2000.3493MATHMathSciNetView ArticleGoogle Scholar
  8. Suzuki T: Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals. Journal of Mathematical Analysis and Applications 2005,305(1):227–239. 10.1016/j.jmaa.2004.11.017MATHMathSciNetView ArticleGoogle Scholar
  9. Petryshyn WV: A characterization of strict convexity of Banach spaces and other uses of duality mappings. Journal of Functional Analysis 1970,6(2):282–291. 10.1016/0022-1236(70)90061-3MATHMathSciNetView ArticleGoogle Scholar
  10. Xu H-K: Another control condition in an iterative method for nonexpansive mappings. Bulletin of the Australian Mathematical Society 2002,65(1):109–113. 10.1017/S0004972700020116MATHMathSciNetView ArticleGoogle Scholar
  11. Minty GJ: On the monotonicity of the gradient of a convex function. Pacific Journal of Mathematics 1964, 14: 243–247.MATHMathSciNetView ArticleGoogle Scholar
  12. Rockafellar RT: Characterization of the subdifferentials of convex functions. Pacific Journal of Mathematics 1966, 17: 497–510.MATHMathSciNetView ArticleGoogle Scholar
  13. Rockafellar RT: On the maximal monotonicity of subdifferential mappings. Pacific Journal of Mathematics 1970, 33: 209–216.MATHMathSciNetView ArticleGoogle Scholar
  14. Rockafellar RT: On the maximality of sums of nonlinear monotone operators. Transactions of the American Mathematical Society 1970,149(1):75–88. 10.1090/S0002-9947-1970-0282272-5MATHMathSciNetView ArticleGoogle Scholar

Copyright

© Y. Yao and R. Chen. 2007

This article is published under license to BioMed Central Ltd. 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.