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Hausdorff continuity of solution maps to equilibrium problems via the oriented distance function

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This paper aims to study the stability in the sense of Hausdorff continuity of solution maps to equilibrium problems without assuming the solid condition of ordered cones. We first propose a generalized concavity of set-valued maps and discuss its relation with the existing concepts. Then, by using the above property and the continuity of the objective function, sufficient conditions for the Hausdorff continuity of solution maps to scalar equilibrium problems are established. Finally, we utilize the oriented distance function to obtain the Hausdorff continuity of solution maps to set-valued equilibrium problems via the corresponding results of the scalar equilibrium problems.

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Acknowledgements

The authors would like to thank the Editors and the anonymous Referees for their helpful remarks and suggestions that helped us significantly improve the paper. This is a result of the project supported by The Ministry of Education and Training of Viet Nam [Grant Number B2023-TCT-05].

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  1. Nguyen Huu Danh and Pham Thanh Duoc have contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, Teacher College, Cantho University, Can Tho, Vietnam

    Lam Quoc Anh

  2. Department of Mathematics, Taydo University, Can Tho, Vietnam

    Nguyen Huu Danh

  3. Faculty of Information Technology, Cantho University of Technology, Can Tho, Vietnam

    Pham Thanh Duoc

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  1. Lam Quoc Anh
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  2. Nguyen Huu Danh
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  3. Pham Thanh Duoc
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Correspondence to Nguyen Huu Danh.

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Quoc Anh, L., Huu Danh, N. & Thanh Duoc, P. Hausdorff continuity of solution maps to equilibrium problems via the oriented distance function. Optim Lett (2024). https://doi.org/10.1007/s11590-024-02098-w

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