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Halpern’s type iterations with perturbations in Hilbert spaces: equilibrium solutions and fixed points

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  • Chih-Sheng Chuang
  • Lai-Jiu Lin
  • Wataru Takahashi

Abstract

In this paper, we consider an iteration process of Halpern’s type for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points for a quasi-nonexpansive mapping with perturbation in a Hilbert space and then prove a strong convergence theorem for such iterations. Using this result, we obtain new strong convergence theorems in a Hilbert space. In particular, we solve partially an open problem posed by Kurokawa and Takahashi (Nonlinear Anal 73:1562–1568, 2010 ) concerning Halpern’s iterations. Copyright Springer Science+Business Media, LLC. 2013

Suggested Citation

  • Chih-Sheng Chuang & Lai-Jiu Lin & Wataru Takahashi, 2013. "Halpern’s type iterations with perturbations in Hilbert spaces: equilibrium solutions and fixed points," Journal of Global Optimization, Springer, vol. 56(4), pages 1591-1601, August.
    • Handle: RePEc:spr:jglopt:v:56:y:2013:i:4:p:1591-1601
    DOI: 10.1007/s10898-012-9911-6
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    References listed on IDEAS

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    1. S. Takahashi & W. Takahashi & M. Toyoda, 2010. "Strong Convergence Theorems for Maximal Monotone Operators with Nonlinear Mappings in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 147(1), pages 27-41, October.
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