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Iterative algorithms for solving fixed point problems and variational inequalities with uniformly continuous monotone operators

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Abstract

Using the double projection and Halpern methods, we prove two strong convergence results for finding a solution of a variational inequality problem involving uniformly continuous monotone operator which is also a fixed point of a quasi-nonexpansive mapping in a real Hilbert space. In our proposed methods, only two projections onto the feasible set in each iteration are performed, rather than one projection for each tentative step during the Armijo-type search, which represents a considerable saving especially when the projection is computationally expensive. We also give some numerical results which show that our proposed algorithms are efficient and implementable from the numerical point of view.

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Acknowledgments

The research was carried out when the first author was an Alexander von Humboldt Postdoctoral Fellow at the Institute of Mathematics, University of Wurzburg, Germany. He is grateful to the Alexander von Humboldt Foundation, Bonn for the fellowship and the Institute of Mathematics, University of Wurzburg, Germany for the hospitality and facilities.

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Authors and Affiliations

  1. Department of Mathematics, University of Nigeria, Nsukka, Nigeria

    Yekini Shehu

  2. Department of Mathematics, Minnesota State University, Moorhead, MN, USA

    Olaniyi S. Iyiola

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  1. Yekini Shehu
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  2. Olaniyi S. Iyiola
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Correspondence to Yekini Shehu.

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The first author is currently an Alexander von Humboldt Postdoctoral Fellow at the Institute of Mathematics, University of Wurzburg, Germany

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Shehu, Y., Iyiola, O.S. Iterative algorithms for solving fixed point problems and variational inequalities with uniformly continuous monotone operators. Numer Algor 79, 529–553 (2018). https://doi.org/10.1007/s11075-017-0449-z

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  • DOI: https://doi.org/10.1007/s11075-017-0449-z

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