Skip to main content
SpringerLink
Log in

Convergence theorems for λ-strict pseudo-contractions in q-uniformly smooth Banach spaces

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

In this paper, we continue to discuss the properties of iterates generated by a strict pseudocontraction or a finite family of strict pseudo-contractions in a real q-uniformly smooth Banach space. The results presented in this paper are interesting extensions and improvements upon those known ones of Marino and Xu [Marino, G., Xu, H. K.: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl., 324, 336–349 (2007)]. In order to get a strong convergence theorem, we modify the normal Mann’s iterative algorithm by using a suitable convex combination of a fixed vector and a sequence in C. This result extends a recent result of Kim and Xu [Kim, T. H., Xu, H. K.: Strong convergence of modified Mann iterations. Nonl. Anal., 61, 51–60 (2005)] both from nonexpansive mappings to λ-strict pseudo-contractions and from Hilbert spaces to q-uniformly smooth Banach spaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Browder, F. E.: Fixed point theorems for noncompact mappings in Hilbert spaces. Prov. Natl. Acad. Sci. USA, 53, 1272–1276 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  2. Browder, F. E.: Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces. Arch. Rational Mech. Anal., 24, 82–90 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  3. Browder, F. E., Petryshyn, W. V.: Construction of fixed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl., 20, 197–228 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  4. Chidume, C. E., Mutangadura, S. A.: An example on the Mann iteration method for Lipschitz pseodocontractions. Proc. Amer. Math. Soc., 129, 2359–2363 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  5. Deimling, K.: Zeros of accretive operators. Manuscripta Math., 13, 365–374 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  6. Opial, Z.: Weak convergence of sequence of successive approximations for nonexpansive mappings. Bull. Amer. Math. Soc., 73, 591–597 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  7. Gossez, J. P., Lami, D. E.: Some geometric properties related to the fixed point theory for nonexpansive mappings. Pacfic J. Math., 40, 565–573 (1972)

    MATH  Google Scholar 

  8. Bruck, R. E.: Nonexpansive projections on subsets of Banach spaces. Pacific J. Math., 47, 341–355 (1973)

    MATH  MathSciNet  Google Scholar 

  9. Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York, 1984

    MATH  Google Scholar 

  10. Reich, S.: Asymptotic behavior of contractions in Banach spaces. J. Math. Anal. Appl., 44, 57–70 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  11. Reich, S.: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. Math. Anal. Appl., 75, 287–292 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  12. Mann, W. R.: Mean value methods in iterations. Proc. Amer. Math. Soc., 14, 506–510 (1953)

    Article  Google Scholar 

  13. Marino, G., Xu, H. K.: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl., 329, 336–349 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Nakajo, K., Takahashi, W.: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl., 279, 372–379 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. Rhoades, B. E.: Fixed point iterations using infinite matrices. Trans. Amer. Math. Soc., 196, 162–176 (1974)

    Article  MathSciNet  Google Scholar 

  16. Zhou, H.: Nonexpansive mappings and iterative methods in uniformly convex Banach spaces. Acta Mathematica Sinica, English Series, 20, 829–836 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  17. Reich, S.: Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl., 75, 274–276 (1979)

    Article  Google Scholar 

  18. Zhou, H.: Convergence theorems of fixed points for κ-strict pseudo-contractions in Hilbert spaces. Nonlinear Anal., 69, 456–462 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  19. Zhou, H.: Convergence theorems for λ-strict pseudo-contractions in 2-uniformly smooth Banach spaces. Nonlinear Anal., 69, 3160–3173 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. Kim, T. H., Xu, H. K.: Strong convergence of modified Mann iterations. Nonlinear Anal., 61, 51–60 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  21. Xu, H. K.: Inequalities in Banach spaces with applications. Nonlinear Anal., 16, 1127–1138 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  22. Zhou, H.: Convergence theorems of common fixed points for a finite family of Lipschitz pseudo-contractions in Banach spaces. Nonlinear Anal., 68, 2977–2983 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  23. Tan, K. K., Xu, H. K.: Fixed point iteration processes for asymptotically nonexpansive mappings. Proc. Amer. Math. Sco., 122, 733–739 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  24. Liu, L. S.: Ishikawa and Mann iteration process with errors for nonlinear strongly accretive mappings in Banach spaces. J. Math. Anal. Appl., 194, 114–125 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  25. Suzuki, T.: Strong convergenve of Krasnoselskii and Mann’s type sequences for one parameter nonexpansive semigroups without Bochner integrals. J. Math. Anal. Appl., 305, 227–239 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  26. Mitrinović, D. S.: Analytic Inequalities, Springer-Verlag, New York, 1970

    MATH  Google Scholar 

  27. Halpern, B.: Fixed points of nonexpansive maps. Bull. Amer. Math. Soc., 73, 957–961 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  28. Lions, P. L.: Approximation de points fixes de contractions. C. R. Acad. Sci. Sér. A-B, 284, 1357–1359 (1977)

    MATH  Google Scholar 

  29. Wittmann, R.: Approximation of fixed points of nonexpansive mappings. Arch. Math., 58, 486–491 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  30. Xu, H. K.: Remarks on an iterative method for nonexpansive mappings. Commun. Appl. Nonlinear Anal., 10, 67–75 (2003)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, 050016, P. R. China

    Hai Yun Zhou

  2. Department of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang, 050003, P. R. China

    Hai Yun Zhou

Authors
  1. Hai Yun Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hai Yun Zhou.

Additional information

Supported by National Natural Science Foundation of China (Grant No. 10771050)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhou, H.Y. Convergence theorems for λ-strict pseudo-contractions in q-uniformly smooth Banach spaces. Acta. Math. Sin.-English Ser. 26, 743–758 (2010). https://doi.org/10.1007/s10114-010-7341-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-010-7341-2

Keywords

MR(2000) Subject Classification

Search

Navigation