Optimal parameter selections for a general Halpern iteration

  • Songnian He
  • Tao Wu
  • Yeol Je ChoEmail author
  • Themistocles M. Rassias
Original Paper


Let C be a closed affine subset of a real Hilbert space H and \(T:C \rightarrow C\) be a nonexpansive mapping. In this paper, for any fixed uC, a general Halpern iteration process:
$$\left\{\begin{array}{ll} x_{0} \in C,\\ x_{n + 1}=t_{n}u+(1-t_{n})Tx_{n},n\geq 0, \end{array}\right. $$
is considered for finding a fixed point of T nearest to u, where the parameter sequence {tn} is selected in the real number field, \(\mathbb {R}\). The core problem to be addressed in this paper is to find the optimal parameter sequence so that this iteration process has the optimal convergence rate and to give some numerical results showing advantages of our algorithms. Also, we study the problem of selecting the optimal parameters for a general viscosity approximation method and apply the results obtained from this study to solve a class of variational inequalities.


Fixed point Nonexpansive mapping Strong convergence Halpern iteration Optimal parameter selection 

Mathematics Subject Classification (2010)

47H09 47H10 65J15 


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Funding information

This work was supported by the Fundamental Research Funds for the Central Universities (3122017078).


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Songnian He
    • 1
  • Tao Wu
    • 1
  • Yeol Je Cho
    • 2
    • 3
    Email author
  • Themistocles M. Rassias
    • 4
  1. 1.College of ScienceCivil Aviation University of ChinaTianjinChina
  2. 2.Department of Mathematics EducationGyeongsang National UniversityJinjuKorea
  3. 3.Center for General EducationChina Medical UniversityTaichungTaiwan
  4. 4.Department of MathematicsNational Technical University of AthensZografou CampusGreece

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