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Retraction algorithms for solving variational inequalities, pseudomonotone equilibrium problems, and fixed-point problems in Banach spaces

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Abstract

In this paper, using sunny generalized nonexpansive retractions which are different from the metric projection and generalized metric projection in Banach spaces, we present new extragradient and line search algorithms for finding the solution of a J-variational inequality whose constraint set is the common elements of the set of fixed points of a family of generalized nonexpansive mappings and the set of solutions of a pseudomonotone J-equilibrium problem for a J -α-inverse-strongly monotone operator in a Banach space. To prove strong convergence of generated iterates in the extragradient method, we introduce a ϕ ∗-Lipschitz-type condition and assume that the equilibrium bifunction satisfies this condition. This condition is unnecessary when the line search method is used instead of the extragradient method. Using FMINCON optimization toolbox in MATLAB, we give some numerical examples and compare them with several existence results in literature to illustrate the usability of our results.

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Acknowledgments

The authors would like to thank the referees for a number of valuable suggestions regarding a previous version of this paper.

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  1. Department of Mathematics, Sahand University of Technology, Tabriz, Iran

    Zeynab Jouymandi & Fridoun Moradlou

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  1. Zeynab Jouymandi
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Correspondence to Fridoun Moradlou.

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Jouymandi, Z., Moradlou, F. Retraction algorithms for solving variational inequalities, pseudomonotone equilibrium problems, and fixed-point problems in Banach spaces. Numer Algor 78, 1153–1182 (2018). https://doi.org/10.1007/s11075-017-0417-7

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