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Epi/Hypo-Convergence of Bifunctions on General Domains and Approximations of Quasi-Variational Problems

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Abstract

Epi/hypo-convergence is extended to the case of bifunctions defined on general domains. Its basic characterizations are established. Variational properties such as those about saddle points, weak saddle points, minsup-points, sup-projections, etc, of bifunctions are shown to be preserved for their epi/hypo-limits (possibly under some additional assumptions). Approximations of quasi-equilibrium problems together with their dual problems in terms of epi/hypo-convergence are considered. The obtained results are new, some are significantly different from the counterparts for the rectangular case and some improve known results, when applied to this special case.

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Acknowledgements

This work was supported by Vietnam National University-Hochiminh City under the grant B2018-28-02. A part of this work was completed during a scientific stay of the authors at Vietnam Institute for Advanced Study in Mathematics (VIASM), whose support and hospitality are grateful. The authors are also very indebted to the associate editor and the referees for their valuable remarks and suggestions for them to improve significantly the first submission.

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Correspondence to P. Q. Khanh.

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Diem, H.T.H., Khanh, P.Q. Epi/Hypo-Convergence of Bifunctions on General Domains and Approximations of Quasi-Variational Problems. Set-Valued Var. Anal (2020). https://doi.org/10.1007/s11228-020-00529-1

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Keywords

  • Epi/hypo-convergence
  • General domains
  • Saddle points
  • Weak saddle points
  • Approximations
  • Quasi-variational problems

Mathematics Subject Classification (2010)

  • 90C31
  • 49J53
  • 47H04