Modified Halpern’s iteration for fixed point theory of a finite family of G-nonexpansive mappings endowed with graph

  • Atid Kangtunyakarn
Original Paper


The aim of this research is to introduce GS-mapping generated by a finite family of G-nonexpansive mappings and finite real numbers and prove a convergence theorem of Halpern iteration associated with GS-mapping for fixed point problem of a finite family of G-nonexpansive mapping in Hilbert spaces endowed with graph, which extends the work of Tiammee et al. (Fixed Point Theory Appl. 2015:187, 2015). Moreover, we introduce a new method for the estimation of value of \(\pi \) using our theorem.


G–S-mapping G-nonexpansive mapping Dominating set 

Mathematics Subject Classification

47H09 47H10 05C69 



This research was supported by Research Administration Division of King Mongkut\(^{,}s\) Institute of Technology Ladkrabang.


  1. 1.
    Halpern, B.: Fixed points of nonexpanding maps. Bull. Am. Math. Soc. 73, 957–961 (1967)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Lions, P.L.: Approximation de points fixes de contractions. (French) C. R. Acad. Sci. Paris Ser. A-B 284, A1357–A1359 (1977)Google Scholar
  3. 3.
    Reich, S.: Approximating fixed points of nonexpansive mappings. Panam. Math. J. 4, 23–28 (1994)MathSciNetMATHGoogle Scholar
  4. 4.
    Wittmann, R.: Approximation of fixed points of nonexpansive mappings. Arch. Math. (Basel) 58, 486–491 (1992)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Kurokawaa, Yu., Takahashi, W.: Weak and strong convergence theorems for nonspreading mappings in Hilbert spaces. Nonlinear Anal. 73, 1562–1568 (2010)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cho, Y.J., Qin, X., Kang, S.M.: Strong convergence of the Modified Halpern-type iterative algorithms in Banach spaces. An. Stiint. Univ. Ovidius Constanta Ser. Mat. 17(1), 51–68 (2009)Google Scholar
  7. 7.
    Qin, X., Su, Y., Shang, M.: Strong convergence of the composite Halpern iteration. J. Math. Anal. Appl. 339, 996–1002 (2008)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Takahashi, W., Wong, N.C., Cho, Y.J.: Attractive points and Halpern-type strong convergence theorems in Hilbert spaces. J. Fixed Point Theory Appl. 17, 301311 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hecai, Y., Min, Z.: Strong convergence of hybrid Halpern processes in a Banach space. J. Nonlinear Sci. Appl. 9, 1776–1786 (2016)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Hu, L.G.: Strong convergence of a modified Halpern’s iteration for nonexpansive mappings. Fixed Point Theory Appl. 2008, Article ID 649162 (2008)Google Scholar
  11. 11.
    Piri, H.: Strong convergence of the CQ method for fixed points of semigroups of nonexpansive mappings. J. Nonlinear Funct. Anal. 2015, Article ID 18 (2015)Google Scholar
  12. 12.
    Qin, X., Cho, S.Y.: Convergence analysis of a monotone projection algorithm in reflexive Banach spaces. Acta Math. Sci. 37B, 488–502 (2017)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kangtunyakarn, A.: Fixed point theory for nonlinear mappings in Banach spaces and applications. Fixed Point Theory Appl. 2014, Article ID 108 (2014)Google Scholar
  14. 14.
    Jachymski, J.: The contraction principle for mappings on a metric space with a graph. Proc. Am. Math. Soc. 136(4), 1359–1373 (2008)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Tiammee, J., Kaewkhao, A., Suantai, S.: On Browder’s convergence theorem and Halpern iteration process for G-nonexpansive mappings in Hilbert spaces endowed with graphs. Fixed Point Theory Appl. 2015, 187 (2015)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kangtunyakarn, A., Suantai, S.: Hybrid iterative scheme for generalized equilibrium problems and fixed point problems of finite family of nonexpansive mappings. Nonlinear Anal. Hybrid Syst. 3, 296–309 (2009)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Liu, L.S.: Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces. J. Math. Anal. Appl. 194, 114–125 (1995)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceKing Mongkut’s Institute of Technology LadkrabangBangkokThailand

Personalised recommendations