On the convergence problems of Ishikawa and Mann iterative processes with error for Φ-pseudo contractive type mappings

  • Zhang Shisheng
Article

Abstract

The purpose of this paper is to introduce the concept of Φ-pseudo contractive type mapping and to study the convergence problem of Ishikawa and Mann iterative processes with error for this kind of mappings. The results presented in this paper improve and extend many authors' recent results.

Key words

Φ-pseudo contractive type mapping accretive mapping pseudo-contractive mapping Φ-strong accretive mapping Φ-hemi-contractive mapping Ishikawa iterative processe with error Mann iterative process with error 

CLC number

O177.91 

References

  1. [1]
    Dunn J C. Iterative construction of fixed points for multivalued operators of the monotone type[J].J. Functi Anal, 1978,27, 38–50.MATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    Chidume C E. Iterative construction of fixed points for multivalued operators of the monotone type [J].Appl Anal, 1996,23:209–218.MathSciNetGoogle Scholar
  3. [3]
    Chang S S, Tan K K. Iteration processes of fixed point for operators of monotone type in Banach spaces[J].Bull Austral Math Soc, 1998,57: 433–445.MATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    Osilike M O. Iterative solution of nonlinear equations of the ϕ-strongly accretive strongly accretive type[J].J Math Anal Appl, 1996,200:259–271.MATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    Browder F E. Nonlinear mappings of nonexpansive and accretive type in Banach spaces [J].Bull Amer Math Soc, 1967,73: 875–882.MATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    Kato T. Nonlinear semi-groups and evolution equations[J].J Math Soc Japan, 1964,19:508–520.CrossRefGoogle Scholar
  7. [7]
    Ishikawa S. Fixed point by a new iteration method[J].Proc Amer Math Soc, 1974,14(1):147–150.MathSciNetCrossRefGoogle Scholar
  8. [8]
    Mann W R. Mean value methods in iteration[J].Proc Amer Math Soc, 1953,4:506–510.MATHMathSciNetCrossRefGoogle Scholar
  9. [9]
    Chang S S. On Chidume's open questions and approximate solutions of multivalued strongly accretive mappings equations in Banach spaces[J].J Math Anal Appl, 1997,216:94–111.MATHMathSciNetCrossRefGoogle Scholar
  10. [10]
    Chang S S. Some problems and results in the study of nonlinear analysis [J].Nonlinear Anal, 1997,30(7):4197–4208.MATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    Chidume C E. Approximation of fixed points of strongly pseudo-contractive mappings[J].Proc Amer Math Soc, 1994,120(2): 545–551.MATHMathSciNetCrossRefGoogle Scholar
  12. [12]
    Chidume C E. Steepest descent approximations for accretive operator equations[J].Nonlinear Anal, 1996,26:299–311.MATHMathSciNetCrossRefGoogle Scholar
  13. [13]
    Chidume C E. Iterative solution of nonlinear equations with strongly accretive operators[J].J Math Anal Appl, 1995,192:502–518.MATHMathSciNetCrossRefGoogle Scholar
  14. [14]
    Chidume C E, Osilike M O. Ishikawa process for nonlinear Lipschitz strongly accretive mappings [J].J Math Anal Appl, 1995,192:727–741.MATHMathSciNetCrossRefGoogle Scholar
  15. [15]
    Deng Lei. On Chidume's open questions[J].J Math Anal Appl, 1993,174(2):441–449.MATHMathSciNetCrossRefGoogle Scholar
  16. [16]
    Deng Lei, Ding Xieping. Iterative approximation of Lipschitz strongly pseudo-contractive mappings in uniformly smooth Banach spaces[J].Nonlinear Anal, 1995,24:981–987.MATHMathSciNetCrossRefGoogle Scholar
  17. [17]
    Osilike M O. Stable iteration procedures of strong pseudo-contractions and nonlinear operator equations of the accretive type[J].J Math Anal Appl, 1996,204:677–692.MATHMathSciNetCrossRefGoogle Scholar
  18. [18]
    Osilike M O. Iterative construction of fixed points of multi-valued operators of the accretive type [J].Soochow J Math, 1996,22(4):485–494.MATHMathSciNetGoogle Scholar
  19. [19]
    Reich S. An iterative procedure for constructing zeros of accretive sets in Banach spaces [J].Nonlinear Anal, 1978,2:85–92.MATHMathSciNetCrossRefGoogle Scholar
  20. [20]
    Reich S. Strong convergence theorems for resolvents of accretive operators in Banach spaces[J].Math Anal Appl, 1980,85:287–292.CrossRefGoogle Scholar
  21. [21]
    Tan K K, Xu H K. Iterative solutions to nonlinear equations of strongly accretive operators in Banach spaces[J].J Math Anal Appl, 1993,178:9–21.MATHMathSciNetCrossRefGoogle Scholar
  22. [22]
    Zhou H Y. Remarks on Ishikawa iteration[J].Chinese Sci Bull, 1977,42:126–128.Google Scholar
  23. [23]
    Liu L S. Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces[J].J Math Anal Appl, 1995,194:114–125.MATHMathSciNetCrossRefGoogle Scholar
  24. [24]
    Zeidler E.Nonlinear Functional Analysis and Its Applications [M]. Part II, Monotone Operators, New York/Berlin: Springer-Verlag, 1985.Google Scholar

Copyright information

© Editorial Committee of Applied Mathematics and Mechanics 2000

Authors and Affiliations

  • Zhang Shisheng
    • 1
  1. 1.Department of MathematicsSichuan UniversityChengduP R China

Personalised recommendations