Abstract
We show that the solution set of a strictly convex combination of equilibrium problems is not necessarily contained in the corresponding intersection of solution sets of equilibrium problems even if the bifunctions defining the equilibrium problems are continuous and monotone. As a consequence, we show that some results given in some recent papers are not always true. Therefore different numerical methods for computing common solutions of families of equilibrium problems proposed in the literature may not converge under the monotonicity assumption. Finally, we prove that if the bifunctions are also parapseudomonotone, then the solution set of any strictly convex combination of a family of equilibrium problems is equivalent to the solution set of the intersection of the same family of equilibrium problems.
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Acknowledgements
The authors would like to thank the editor in chief, the editors and the referees very much for their constructive comments and suggestions, especially on the presentation and the structure of their submitted version. These helped them very much in revising their paper.
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This paper is dedicated to Professor Le Dung Muu on the Occasion of His 70th Birthday.
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Ha, N.T.T., Thanh, T.T.H., Hai, N.N. et al. A note on the combination of equilibrium problems. Math Meth Oper Res 91, 311–323 (2020). https://doi.org/10.1007/s00186-019-00690-w
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DOI: https://doi.org/10.1007/s00186-019-00690-w