Skip to content


  • Research Article
  • Open Access

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

Fixed Point Theory and Applications20072007:032870

  • Received: 11 October 2006
  • Accepted: 11 December 2006
  • Published:


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.


  • Hilbert Space
  • Approximate Solution
  • Variational Inequality
  • Convex Function
  • Minimization Problem


Authors’ Affiliations

Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300160, China


  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