Journal of Applied Mathematics and Computing

, Volume 32, Issue 2, pp 453–464 | Cite as

Convergence theorems of a modified hybrid algorithm for a family of quasi-φ-asymptotically nonexpansive mappings

Article
First Online:
Received:
Revised:

Abstract

The purpose of this article is to propose a modified hybrid projection algorithm and prove strong convergence theorems for a family of quasi-φ-asymptotically nonexpansive mappings. The results of this paper improve and extend the results of S. Matsushita and W. Takahashi (J. Approx. Theory, 134: 257–266 (2005)), T.H. Kim, H.K. Xu (Nonlinear Anal. 64: 1140–1152 (2006)), Y.F. Su, D.X. Wang, M.J. Shang (Fixed Point Theory Appl. 2008: 284613 (2008)) and others.

Keywords

Closed mapping Quasi-φ-asymptotically nonexpansive mapping Asymptotically regular mapping Generalized projection Modified hybrid projection iteration algorithm Strong convergence theorem 

Mathematics Subject Classification (2000)

47H09 47H10 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Goebel, K., Kirk, W.A.: A fixed point theorem for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 35, 171–174 (1972) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Schu, J.: Iteration construction of fixed points of asymptotically nonexpansive mappings. J. Math. Anal. Appl. 158, 407–413 (1991) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Zhou, H., Cho, Y.J., Kang, S.M.: A new iterative algorithm for approximating common fixed points for asymptotically nonexpansive mappings. Fixed Point Theory Appl. 2007, 64874 (2007) CrossRefMathSciNetGoogle Scholar
  4. 4.
    Haugazeau, Y.: Sur les inéquations variationnelles et la minimisation de fonctionnelles convexes. Thése, Université de Paris, Paris, France Google Scholar
  5. 5.
    Bauschke, H.H., Combettes, P.L.: A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces. Math. Oper. Res. 26, 248–264 (2001) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Nakajo, K., Takahashi, W.: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl. 279, 372–379 (2003) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Kim, T.H., Xu, H.K.: Strong convergence of modified Mann iterations for asymptotically mappings and semigroups. Nonlinear Anal. 64, 1140–1152 (2006) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Carlos, M.Y., Xu, H.K.: Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Anal. 64, 2240–2411 (2006) Google Scholar
  9. 9.
    Matsushita, S.Y., Takahashi, W.: A strong convergence theorem for relatively nonexpansive mappings in a Banach space. J. Approx. Theory 134, 257–266 (2005) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Alber, Ya.I.: Metric and generalized projection operators in Banach spaces: properties and applications. In: Kartsatos, A.G. (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, pp. 15–50. Dekker, New York (1996) Google Scholar
  11. 11.
    Alber, Ya.I., Reich, S.: An iterative method for solving a class of nonlinear operator equations in Banach spaces. Panam. Math. J 4, 39–54 (1994) MATHMathSciNetGoogle Scholar
  12. 12.
    Kamimura, S., Takahashi, W.: Strong convergence of a proximal-type algorithm in a Banach space. SIAM J. Optim. 13, 938–945 (2002) CrossRefMathSciNetGoogle Scholar
  13. 13.
    Cioranescu, I.: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic, Dordrecht (1990) MATHGoogle Scholar
  14. 14.
    Takahashi, W.: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000) MATHGoogle Scholar
  15. 15.
    Su, Y.F., Qin, X.: Strong convergence of modified Ishikawa iterations for nonlinear mappings. Proc. Indian Acad. Sci., Math. Sci. 117, 97–107 (2007) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Su, Y.F., Wang, D.X., Shang, M.J.: Strong convergence of monotone hybrid algorithm for hemi-relatively nonexpansive mappings. Fixed Point Theory Appl. 2008, 284613 (2008) CrossRefMathSciNetGoogle Scholar
  17. 17.
    He, H.M., Chen, R.D.: Strong convergence theorems of the CQ method for nonexpansive mappings. Fixed Point Theory Appl. 2007, 59735 (2007) CrossRefMathSciNetGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2009

Authors and Affiliations

  1. 1.Department of MathematicsShijiazhuang Mechanical Engineering CollegeShijiazhuangP.R. China

Personalised recommendations