A New Simple Parallel Iteration Method for a Class of Variational Inequalities

Abstract

In this paper, we propose a new simple parallel iterative method to find a solution for variational inequalities over the set of common fixed points of an infinite family of nonexpansive mappings on real reflexive and strictly convex Banach spaces with a uniformly Gâteaux differentiable norm. Our parallel iterative method is simpler than the one proposed by Buong et al. (Numer. Algorithms 72, 467–481 2016). An iterative method of Halpern type for common zeros of an infinite family of m-accretive mappings is shown as a special case of our result. Two numerical examples are also given to illustrate the effectiveness and superiority of the proposed algorithm.

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Acknowledgements

The authors are extremely grateful to Prof. Nguyen Tu Cuong and the referees for many helpful comments and suggestions, which helped to improve this paper.

Funding

This work was supported by the Vietnam National Foundation for Science and Technology Development under the code 101.02-2017.305.

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Correspondence to Nguyen Song Song Ha.

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Ha, N.S., Buong, N. & Thuy, N.T.T. A New Simple Parallel Iteration Method for a Class of Variational Inequalities. Acta Math Vietnam 43, 239–255 (2018). https://doi.org/10.1007/s40306-017-0228-x

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Keywords

  • Fixed point
  • Nonexpansive and accretive mapping
  • Variational inequality
  • Common zero of accretive mappings
  • Iterative algorithms

Mathematics Subject Classification (2010)

  • 47J05
  • 47H09
  • 49J30