Skip to content


  • Research Article
  • Open Access

Weak convergence of an iterative sequence for accretive operators in Banach spaces

Fixed Point Theory and Applications20062006:35390

  • Received: 21 November 2005
  • Accepted: 6 December 2005
  • Published:


Let be a nonempty closed convex subset of a smooth Banach space and let be an accretive operator of into . We first introduce the problem of finding a point such that where is the duality mapping of . Next we study a weak convergence theorem for accretive operators in Banach spaces. This theorem extends the result by Gol'shteĭn and Tret'yakov in the Euclidean space to a Banach space. And using our theorem, we consider the problem of finding a fixed point of a strictly pseudocontractive mapping in a Banach space and so on.


  • Banach Space
  • Differential Geometry
  • Weak Convergence
  • Computational Biology
  • Iterative Sequence


Authors’ Affiliations

Department of Economics, Chiba University, Yayoi-Cho, Inage-Ku,Chiba-Shi, Chiba 263-8522, Japan
Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Oh-Okayama, Meguro-Ku, Tokyo 152-8522, Japan


  1. Ball K, Carlen EA, Lieb EH: Sharp uniform convexity and smoothness inequalities for trace norms. Inventiones Mathematicae 1994,115(3):463–482.MathSciNetView ArticleMATHGoogle Scholar
  2. Beauzamy B: Introduction to Banach Spaces and Their Geometry, North-Holland Mathematics Studies. Volume 68. 2nd edition. North-Holland, Amsterdam; 1985:xv+338.Google Scholar
  3. Brezis H: Analyse Fonctionnelle. Théorie et Applications, Collection of Applied Mathematics for the Master's Degree. Masson, Paris; 1983:xiv+234.Google Scholar
  4. Browder FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. In Nonlinear Functional Analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill, 1968). American Mathematical Society, Rhode Island; 1976:1–308.Google Scholar
  5. Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert space. Journal of Mathematical Analysis and Applications 1967, 20: 197–228. 10.1016/0022-247X(67)90085-6MathSciNetView ArticleMATHGoogle Scholar
  6. Bruck RE Jr.: Nonexpansive retracts of Banach spaces. Bulletin of the American Mathematical Society 1970, 76: 384–386. 10.1090/S0002-9904-1970-12486-7MathSciNetView ArticleMATHGoogle Scholar
  7. Bruck RE Jr.: A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces. Israel Journal of Mathematics 1979,32(2–3):107–116. 10.1007/BF02764907MathSciNetView ArticleMATHGoogle Scholar
  8. Gol'shteĭn EG, Tret'yakov NV: Modified Lagrangians in convex programming and their generalizations. Mathematical Programming Study 1979, (10):86–97.Google Scholar
  9. Iiduka H, Takahashi W: Strong convergence of a projection algorithm by hybrid type for monotone variational inequalities in a Banach space. in preparationGoogle Scholar
  10. Iiduka H, Takahashi W: Weak convergence of a projection algorithm for variational inequalities in a Banach space. in preparationGoogle Scholar
  11. Iiduka H, Takahashi W, Toyoda M: Approximation of solutions of variational inequalities for monotone mappings. Panamerican Mathematical Journal 2004,14(2):49–61.MathSciNetMATHGoogle Scholar
  12. Kamimura S, Takahashi W: Weak and strong convergence of solutions to accretive operator inclusions and applications. Set-Valued Analysis 2000,8(4):361–374. 10.1023/A:1026592623460MathSciNetView ArticleMATHGoogle Scholar
  13. Kinderlehrer D, Stampacchia G: An Introduction to Variational Inequalities and Their Applications, Pure and Applied Mathematics. Volume 88. Academic Press, New York; 1980:xiv+313.MATHGoogle Scholar
  14. Kirk WA: A fixed point theorem for mappings which do not increase distances. The American Mathematical Monthly 1965, 72: 1004–1006. 10.2307/2313345MathSciNetView ArticleMATHGoogle Scholar
  15. Kitahara S, Takahashi W: Image recovery by convex combinations of sunny nonexpansive retractions. Topological Methods in Nonlinear Analysis 1993,2(2):333–342.MathSciNetMATHGoogle Scholar
  16. Lau AT, Takahashi W: Weak convergence and nonlinear ergodic theorems for reversible semigroups of nonexpansive mappings. Pacific Journal of Mathematics 1987,126(2):277–294.MathSciNetView ArticleMATHGoogle Scholar
  17. Lions J-L, Stampacchia G: Variational inequalities. Communications on Pure and Applied Mathematics 1967, 20: 493–519. 10.1002/cpa.3160200302MathSciNetView ArticleMATHGoogle Scholar
  18. Liu F, Nashed MZ: Regularization of nonlinear ill-posed variational inequalities and convergence rates. Set-Valued Analysis 1998,6(4):313–344. 10.1023/A:1008643727926MathSciNetView ArticleMATHGoogle Scholar
  19. Osilike MO, Udomene A: Demiclosedness principle and convergence theorems for strictly pseudocontractive mappings of Browder-Petryshyn type. Journal of Mathematical Analysis and Applications 2001,256(2):431–445. 10.1006/jmaa.2000.7257MathSciNetView ArticleMATHGoogle Scholar
  20. Reich S: Asymptotic behavior of contractions in Banach spaces. Journal of Mathematical Analysis and Applications 1973,44(1):57–70. 10.1016/0022-247X(73)90024-3MathSciNetView ArticleMATHGoogle Scholar
  21. Reich S: Weak convergence theorems for nonexpansive mappings in Banach spaces. Journal of Mathematical Analysis and Applications 1979,67(2):274–276. 10.1016/0022-247X(79)90024-6MathSciNetView ArticleMATHGoogle Scholar
  22. Takahashi W: Nonlinear Functional Analysis. Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama; 2000:iv+276.MATHGoogle Scholar
  23. Takahashi Y, Hashimoto K, Kato M: On sharp uniform convexity, smoothness, and strong type, cotype inequalities. Journal of Nonlinear and Convex Analysis 2002,3(2):267–281.MathSciNetMATHGoogle Scholar
  24. Takahashi W, Kim G-E: Approximating fixed points of nonexpansive mappings in Banach spaces. Mathematica Japonica 1998,48(1):1–9.MathSciNetMATHGoogle Scholar
  25. Xu HK: Inequalities in Banach spaces with applications. Nonlinear Analysis 1991,16(12):1127–1138. 10.1016/0362-546X(91)90200-KMathSciNetView ArticleMATHGoogle Scholar