Applied Mathematics and Mechanics

, Volume 28, Issue 10, pp 1287–1297 | Cite as

Strong convergence theorems for nonexpansive semi-groups in Banach spaces

  • Zhang Shi-sheng  (张石生)
  • Yang Li  (杨莉)
  • Liu Jing-ai  (刘京爱)
Article

Abstract

Some strong convergence theorems of explicit composite iteration scheme for nonexpansive semi-groups in the framework of Banach spaces are established. Results presented in the paper not only extend and improve the corresponding results of Shioji-Takahashi, Suzuki, Xu and Aleyner-Reich, but also give a partially affirmative answer to the open questions raised by Suzuki and Xu.

Key words

nonexpansive semi-group demi-closed principle common fixed point uniformly smooth Banach space weakly continuous normalized duality mapping 

Chinese Library Classification

O177.91 

2000 Mathematics Subject Classification

47H20 47H10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Browder F E. Fixed point theorems for noncompact mappings in Hilbert space[J]. Proc Nat Acad Sci U S A, 1965, 53:1272–1276.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Reich S. Strong convergence theorems for resolvents of accretive operators in Banach spaces[J]. J Math Anal Appl, 1980, 75:287–292.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    Shioji N, Takahashi W. Strong convergence theorems for asymptotically nonexpansive mappings in Hilbert spaces[J]. Nonlinear Anal, 1998, 34:87–99.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Suzuki T. On strong convergence to a common fixed point of nonexpansive semi-group in Hilbert spaces[J]. Proc Amer Math Soc, 2003, 131(7): 2133–2136.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Xu H K. A strong convergence theorem for contraction semi-groups in Banach spaces[J]. Bull Austral Math Soc, 2005, 72:371–379.MATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    Aleyner A, Reich S. An explicit construction of sunny nonexpansive retractions in Banach spaces[J]. Fixed Point Theory and Applications, 2005, 3:295–305.CrossRefMathSciNetGoogle Scholar
  7. [7]
    Xu H K. Strong convergence of an iterative method for nonexpansive and accretive operators[J]. J Math Anal Appl, 2006, 314:631–643.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    Goebel K, Kirk W A. Topics in metric fixed point theory[M]. In: Cambridge Studies in Advanced Mathematics, Cambridge: Cambridge Univ Press, 1990.Google Scholar
  9. [9]
    Barbu V. Nonlinear semi-groups and differential equations in Banach spaces[M]. Noordhoff, 1976.Google Scholar
  10. [10]
    Bruck R E. Nonexpansive projections on subsets of Banach spaces[J]. Pacific J Math, 1973, 47:341–355.MATHMathSciNetGoogle Scholar
  11. [11]
    Reich S. Asymptotic behavior of contractions in Banach spaces[J]. J Math Anal Appl, 1973, 44:57–70.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    Browder F E. Convergence theorems for sequences of nonlinear operators in Banach spaces[J]. Math Z, 1967, 100:201–225.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    Goebel K, Reich S. Uniform convexity, hyperbolic geometry and nonexpansive mappings[M]. In: Monographs and Textbooks in Pure and Applied Mathematics, New York: Marcel Dekker Inc, 1984.Google Scholar
  14. [14]
    Liu L S. Ishikawa and Mann iterarive processes with errors for nonlinear strongly accretive mappings in Banach space[J]. J Math Anal Appl, 1995, 194:114–125.MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Chang S S. On Chidume’s open questions and approximation solutions of multi-valued strongly accretive mappings equations in Banach spaces[J]. J Math Anal Appl, 1977, 216:94–111.CrossRefGoogle Scholar

Copyright information

© Editorial Committee of Appl. Math. Mech. 2007

Authors and Affiliations

  • Zhang Shi-sheng  (张石生)
    • 1
  • Yang Li  (杨莉)
    • 2
  • Liu Jing-ai  (刘京爱)
    • 3
  1. 1.Department of MathematicsYibin UniversityYibinP. R. China
  2. 2.Department of MathematicsSouth West University of Science and TechnologyMianyangP. R. China
  3. 3.Department of MathematicsYanbian UniversityYanjiP. R. China

Personalised recommendations