Advertisement

Numerical Algorithms

, Volume 81, Issue 3, pp 1129–1148 | Cite as

Approximating fixed points of generalized α-nonexpansive mappings in Banach spaces by new faster iteration process

  • H. PiriEmail author
  • B. Daraby
  • S. Rahrovi
  • M. Ghasemi
Original Paper
  • 95 Downloads

Abstract

In this paper, we introduce a new iterative scheme to approximate fixed point of generalized α-nonexpansive mappings and then, we prove that the proposed iteration process is faster than all of Picard, Mann, Ishikawa, Noor, Agarwal, Abbas, and Thakur processes for contractive mappings. We also obtain some weak and strong convergence theorems for generalized α-nonexpansive mappings. At the end, by using an example for generalized α-nonexpansive mappings, we compare the convergence behavior of new iterative process with other iterative processes.

Keywords

Uniformly convex Banach space Convergence theorem Generalized α-nonexpansive mapping Opial’s property 

Mathematics Subject Classification (2010)

Primary: 47H05 Secondary: 47H09, 47H20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Suzuki, T.: Fixed point theorems and convergence theorems for some generalized nonexpansive mappings. J. Math. Anal Appl. 340, 1088–1095 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aoyama, K., Kohsaka, F.: Fixed point theorem for α-nonexpansive mappings in Banach spaces. Nonlinear Anal. 74, 4387–4391 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ariza-Ruiz, D., Hermandez Linares, C., Llorens-Fuster, E., Moreno-Galvez, E.: On α-nonexpansive mappings in Banach spaces. Carpath. J. Math. 32, 13–28 (2016)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Pant, R., Shukla, R.: Approximating fixed points of generalized α-nonexpansive mappings in Banach spaces. Numer. Funct. Anal. Optim. 38, 248–266 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Mann, W. R.: Mean value methods in iteration. Proc. Amer. Math. Soc. 4, 506–510 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ishikawa, S.: Fixed points by a new iteration method. Proc. Amer. Math. Soc. 44, 147–150 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Noor, M. A.: New approximation schemes for general variational inequalities. J. Math. Anal. Appl. 251, 217–229 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Agarwal, R.P., O’Regan, D., Sahu, D.R.: Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex Anal. 8, 61–79 (2007)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Abbas, M., Nazir, T.: A new faster iteration process applied to constrained minimization and feasibility problems. Mat. Vesn. 66, 223–234 (2014)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Shahin, F., Adrian, G., Mihai, P., Shahram, R.: A comparative study on the convergence rate of some iteration methods involving contractive mappings. Fixed Point Theory Appl, pp. 24 (2015)Google Scholar
  11. 11.
    Thakur, B.S., Thakur, D., Postolache, M.: A new iterative scheme for numerical reckoning fixed points of Suzuki’s generalized nonexpansive mappings. App. Math. Comp. 275, 147–155 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Amer. Math. Soc. 73, 591–597 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Takahashi, W.: Nonlinear Functional Analysis. Yokohoma Publishers, Yokohoma (2000)zbMATHGoogle Scholar
  14. 14.
    Agarwal, R.P., O’Regan, D., Sahu, D.R.: Fixed Point Theory for Lipschitzian-type Mappings with Applications Series: Topological Fixed Point Theory and Its Applications, vol. 6. Springer, New York (2009)zbMATHGoogle Scholar
  15. 15.
    Schu, J.: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Austral. Math. Soc. 43, 153–159 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Sahu, D.R.: Applications of the S-iteration process to constrained minimization problems and split feasibility problems. Fixed Point Theory 12, 187–204 (2011)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Berinde, V.: Picard iteration converges faster than Mann iteration for a class of quasi contractive operators. Fixed Point Theory Appl. 2, 97–105 (2004)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BonabBonabIran
  2. 2.Department of MathematicsUniversity of MaraghehMaraghehIran

Personalised recommendations