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Extragradient Methods and Linesearch Algorithms for Solving Ky Fan Inequalities and Fixed Point Problems

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Abstract

In this paper, we introduce some new iterative methods for finding a common element of the set of points satisfying a Ky Fan inequality, and the set of fixed points of a contraction mapping in a Hilbert space. The strong convergence of the iterates generated by each method is obtained thanks to a hybrid projection method, under the assumptions that the fixed-point mapping is a ξ-strict pseudocontraction, and the function associated with the Ky Fan inequality is pseudomonotone and weakly continuous. A Lipschitz-type condition is assumed to hold on this function when the basic iteration comes from the extragradient method. This assumption is unnecessary when an Armijo backtracking linesearch is incorporated in the extragradient method. The particular case of variational inequality problems is examined in a last section.

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Acknowledgements

This research was supported by the Institute for Computational Science and Technology at Ho Chi Minh City (ICST HCMC), Vietnam.

The authors thank the Editor, the Associate Editor, and two referees for their helpful suggestions for improving this paper.

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Correspondence to Jean Jacques Strodiot.

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Communicated by Jean-Pierre Crouzeix.

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Vuong, P.T., Strodiot, J.J. & Nguyen, V.H. Extragradient Methods and Linesearch Algorithms for Solving Ky Fan Inequalities and Fixed Point Problems. J Optim Theory Appl 155, 605–627 (2012). https://doi.org/10.1007/s10957-012-0085-7

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  • DOI: https://doi.org/10.1007/s10957-012-0085-7

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