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On Hölder Calmness and Hölder Well-Posedness of Vector Quasi-Equilibrium Problems

Abstract

We study stability and well-posedness of parametric vector quasi-equilibrium problems. Weak continuity of orders not greater than one around a given point, in the sense of Hölder calmness of such orders, of solution maps is under consideration. Namely, we consider stability in terms of Hölder calmness of solution maps at the considered point of parameter. Sufficient conditions for such Hölder calmness are established for weak and strong vector quasi-equilibrium problems. When applied to the particular case of scalar equilibrium problems, our results recover recent ones appearing online first in the literature. Then we propose a Hölder well-posedness notion for parametric vector quasi-equilibrium problems, based on Hölder calmness of approximate solution maps, and derive sufficient conditions for Hölder well-posedness of both the mentioned weak and strong vector quasi-equilibrium problems.

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Acknowledgements

This research was funded by Vietnam National University Hochiminh City (VNU-HCM) under grant number B2013-28-01. The authors would like to thank the anonymous referee for his/her valuable remarks and suggestions, which have helped them to improve the paper.

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Correspondence to Phan Quoc Khanh.

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Anh, L.Q., Khanh, P.Q., Tam, T.N. et al. On Hölder Calmness and Hölder Well-Posedness of Vector Quasi-Equilibrium Problems. Viet J Math 41, 507–517 (2013). https://doi.org/10.1007/s10013-013-0039-x

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  • DOI: https://doi.org/10.1007/s10013-013-0039-x

Keywords

  • Vector quasi-equilibrium problems
  • Hölder calmness
  • Relaxed Hölder monotonicity
  • Hölder well-posedness

Mathematics Subject Classification (2010)

  • 49K40
  • 90C31
  • 91B50
  • 49J40