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New extragradient methods for solving variational inequality problems and fixed point problems

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Abstract

In this paper, we introduce two new iterative algorithms for finding a common element of the set of fixed points of a quasi-nonexpansive mapping and the set of solutions of the variational inequality problem with a monotone and Lipschitz continuous mapping in real Hilbert spaces, by combining a modified Tseng’s extragradient scheme with the Mann approximation method. We prove weak and strong convergence theorems for the sequences generated by these iterative algorithms. The main advantages of our algorithms are that the construction of solution approximations and the proof of convergence of the algorithms are performed without the prior knowledge of the Lipschitz constant of cost operators. Finally, we provide numerical experiments to show the efficiency and advantage of the proposed algorithms.

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Acknowledgements

The authors would like to thank Professor Simeon Reich and the referee(s) for their comments on the manuscript which helped us very much in improving and presenting the original version of this paper. The second author was partially supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under the project 101.01-2017.315.

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Authors and Affiliations

  1. Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Duong Viet Thong

  2. Department of Mathematics, College of Air Force, Nhatrang City, Vietnam

    Dang Van Hieu

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  1. Duong Viet Thong
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  2. Dang Van Hieu
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Correspondence to Duong Viet Thong.

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Thong, D.V., Van Hieu, D. New extragradient methods for solving variational inequality problems and fixed point problems. J. Fixed Point Theory Appl. 20, 129 (2018). https://doi.org/10.1007/s11784-018-0610-x

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