Lobachevskii Journal of Mathematics

, Volume 31, Issue 1, pp 18–26 | Cite as

Weak and strong convergence theorems for a finite family of non-lipschitzian mappings in Banach spaces

Article
  • 40 Downloads

Abstract

The purpose of this paper is to study a new iterative scheme for a finite family of asymptotically nonexpansive mappings in the intermediate sense. Weak and strong convergence theorems for the iterative scheme in a uniformly convex Banach space are established under some conditions which are weaker than demicompactness or completely continuous. Our results improve and generalize the recent known results in the literature.

Key words and phrases

Strong and weak convergence Asymptotically nonexpansive mapping in the intermediate sense Uniformly convex Opial’s property 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R. E. Bruck, T. Kuczumow, and S. Reich, Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform opial property, Colloq. Math. 65, 169 (1993).MATHMathSciNetGoogle Scholar
  2. 2.
    C. E. Chidume and Bashir Ali, Weak and strong convergence theorems for finite families of asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 330, 377 (2007).MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35, 171 (1972).MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    S. Imnang and S. Suantai, A new iterative method for common fixed points of a finite family of nonexpansive mappings, International Journal of Mathematics and Mathematical Sciences (2009), doi:10.1155/2009/391839.Google Scholar
  5. 5.
    G. Kim and T. Kim, Mann and Ishikawa iterations with errors for non-Lipschitzian mappings in Banach spaces, Computers and Mathematics with Applications. 42, 1565 (2001).MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    W. A. Kirk, Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type, Israel J. Math. 17, 339 (1974).MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    K. Nammanee and S. Suantai, The modified Noor iterations with errors for non-Lipschitzian mappings in Banach spaces, Appl. Math. Comput. 187, 669 (2007).MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Z. Opial, Weak convergence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73, 591 (1967).MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    M. O. Osilike and S. C. Aniagbosor, Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings, Mathematical and Computer Modelling 32, 1181 (2000).MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    B. E. Rhoades, Fixed point iterations for certain nonlinear mappings, J. Math. Anal. Appl. 183, 118 (1994).MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    J. Schu, Iterative construction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal. Appl. 18-2, 407 (1991).CrossRefMathSciNetGoogle Scholar
  12. 12.
    J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc. 43, 153 (1991).MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    H. F. Senter and W. G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. soc. 44(2), 375 (1974).MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    S. Suantai, Weak and strong convergence Criteria of Noor Iterations for asymptotically nonexpansive mappings, J. Math. Anal. Appl. 311, 506 (2005).MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    B. L. Xu and M. A. Noor, Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 267, 444 (2002).MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    L. Yang, X. Xie, and G. Hu, Demi-closed principle and convergence for modified three step iterative process with errors of non-Lipschitzian mappings, Journal of Computational and Applied Mathematics (2009), doi:10.1016/j.cam.2009.01.022.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceThaksin University Phatthalung CampusPhatthalungThailand
  2. 2.Department of Mathematics, Faculty of ScienceChiang Mai UniversityChiang MaiThailand

Personalised recommendations