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Painlevé–Kuratowski convergences of the solution sets for generalized vector quasi-equilibrium problems

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Abstract

In this paper, we consider vector quasi-equilibrium problems under perturbation in terms of suitable asymptotically solving sequences, not embedding given problems into a parameterized family. By employing some types of convergences for mapping and set sequences, we obtain the Painlevé–Kuratowski upper convergence of solution sets for the reference problems. Then, using nonlinear scalarization functions, we propose gap functions for such problems, and later employing these functions, we study necessary and sufficient conditions for Painlevé–Kuratowski lower convergence and Painlevé–Kuratowski convergence. As an application, we discuss the special case of vector quasi-variational inequality.

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Acknowledgements

The authors wish to thank the referees for their valuable remarks and suggestions that helped to improve the presentation of the paper. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.01-2017.18.

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Authors and Affiliations

  1. Department of Mathematics, Teacher College, Can Tho University, Can Tho, Viet Nam

    Lam Quoc Anh

  2. Valaya Alongkorn Rajabhat University under the Royal Patronage, Pathumtani, Thailand

    Thanatporn Bantaojai

  3. Department for Management of Science and Technology Development, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Nguyen Van Hung

  4. Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Nguyen Van Hung

  5. Department of Mathematics, Dong Thap University, 783 Pham Huu Lau Street, Ward 6, Cao Lanh City, Vietnam

    Vo Minh Tam

  6. Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok, 65000, Thailand

    Rabian Wangkeeree

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  1. Lam Quoc Anh
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  2. Thanatporn Bantaojai
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  3. Nguyen Van Hung
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  4. Vo Minh Tam
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  5. Rabian Wangkeeree
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Corresponding author

Correspondence to Nguyen Van Hung.

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Communicated by Carlos Conca.

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Anh, L.Q., Bantaojai, T., Van Hung, N. et al. Painlevé–Kuratowski convergences of the solution sets for generalized vector quasi-equilibrium problems. Comp. Appl. Math. 37, 3832–3845 (2018). https://doi.org/10.1007/s40314-017-0548-4

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  • DOI: https://doi.org/10.1007/s40314-017-0548-4

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