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Extragradient-Proximal Methods for Split Equilibrium and Fixed Point Problems in Hilbert Spaces

Abstract

In this paper, we propose two new extragradient-proximal algorithms for solving split equilibrium and fixed point problems (SEFPP) in real Hilbert spaces, in which the first equilibrium bifunction is pseudomonotone, the second one is monotone, and the fixed point mappings are nonexpansive. By using the extragradient method incorporated with the proximal point algorithm and cutting techniques, we obtain algorithms for solving (SEFPP). Under certain conditions on parameters, the iteration sequences generated by the proposed algorithms are proved to be weakly and strongly convergent to a solution of (SEFPP). Our results improve and extend the previous results given in the literature.

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Acknowledgments

The authors would like to thank the referees very much for their constructive comments and suggestions, especially on the presenting and the structure of the early version of their paper which helped them very much in revising the paper. Their thanks would be addressed to Prof. Le Dung Muu and Prof. Pham Ky Anh for the guidance and discussion. The first author is supported in part by NAFOSTED, under the project 101.01-2014-24, and a grant from Le Quy Don Technical University.

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Correspondence to Bui Van Dinh.

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Van Dinh, B., Son, D.X. & Anh, T.V. Extragradient-Proximal Methods for Split Equilibrium and Fixed Point Problems in Hilbert Spaces. Vietnam J. Math. 45, 651–668 (2017). https://doi.org/10.1007/s10013-016-0237-4

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Keywords

  • Split equilibrium problem
  • Split fixed point problem
  • Nonexpansive mapping
  • Weak and strong convergence
  • Pseudomonotonicity

Mathematics Subject Classification (2010)

  • 47H09
  • 47J25
  • 65K10
  • 65K15
  • 90C99