Abstract
Epi/hypo convergence of finite-valued bivariate functions defined on the product of two subsets, with some connections to lopsided convergence, is considered. Namely, we deal with three full characterizations of this convergence: by epi/hypo convergence of the corresponding proper bifunctions, by explicit formulae of the lower and upper members of the intervals of the limits, and by the bicontinuity of the partial Legendre–Fenchel transform (i.e., the (extended) epi/hypo convergence of bifunctions is characterized by the epi-convergence of their convex parents). We emphasize that epi/hypo limits are not unique and form an entire equivalence class.
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Acknowledgments
This research was supported by the National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2014.62. The second author is very grateful to Professor Roger J-B Wets for the suggestion of the topic and many important discussions during his work. He appreciates also Professor Alejandro Jofré for helpful discussions. The authors are deeply indebted to the anonymous reviewer for the valuable remarks and suggestions, which helped them so much in completing the final version of this paper.
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Dedicated to Professor Nguyen Khoa Son on the occasion of his 65th birthday
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Diem, H.T.H., Khanh, P.Q. Criteria for Epi/Hypo Convergence of Finite-Valued Bifunctions. Vietnam J. Math. 43, 439–458 (2015). https://doi.org/10.1007/s10013-015-0139-x
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DOI: https://doi.org/10.1007/s10013-015-0139-x
Keywords
- Epi-convergence
- Epi/hypo convergence
- Lopsided convergence
- Equivalence class
- Partial Legendre–Fenchel transform
- Partial skew conjugate