Abstract
Motivated by some deep problems in optimization and control theory, convexity theory has been extended to the various infinite dimensional functional spaces. The separation theorems play key roles in develo** convexity theory. In this paper, we focus on the convexity problem in the framework of fuzzy quasi-normed spaces. First, we give some separation results of convex sets, and show a characterization of a closed convex subset with the aid of the dual of a fuzzy quasi-normed space. Second, we introduce the concept of an extreme point and generalize the famous Krein–Milman theorem to a fuzzy quasi-normed space. The obtained results of this paper will play key roles in convex programming and optimization problems of fuzzy quasi-normed spaces.
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References
Alegre C, Romaguera S (2003) On paratopological vector spaces. Acta Math Hungar 101:237–261
Alegre C, Romaguera S (2010) Characterizations of metrizable topological vector spaces and their asymmetric generalizations in terms of fuzzy (quasi-)norms. Fuzzy Sets Syst 161:2181–2192
Alegre C, Romaguera S (2016) On the uniform bundedness theorem in fuzzy quasi-normed spaces. Fuzzy Sets Syst 282:143–153
Bag T, Samanta SK (2003) Finite dimensional fuzzy normed linear spaces. J Fuzzy Math 11:687–705
Cheng SC, Mordeson JN (1994) Fuzzy linear operators and fuzzy normed linear spaces. Bull Calcutta Math Soc 86:429–436
Cobzas S (2013) Functional analysis in asymmetric normed spaces. Springer, Basel
Gao R, Li XX, Wu JR (2020) The decomposition theorem for a fuzzy quasi-norm. J Math 2020:1–7
Holmes RB (1975) Geometrical functional analysis and its applications. Springer, Berlin
Hussein BY, Al-Basri FK (2020) On the completion of quasi-fuzzy normed algebra over fuzzy field. J Interdip Math 25:1–13
Ioffe AD, Tikhomirov VM (1979) Theory of Extremal Problems [Russian]. Nauka, Moscow. (English translation: North-Holland, Amsterdam-Oxford)
Katsaras AK (1984) Fuzzy topological vector spaces II. Fuzzy Sets Syst 12:143–154
Li RN, Wu JR (2022) Hahn-Banach type theorems and the separation of convex sets for fuzzy quasi-normed spaces. AIMS Math 7(3):3290–3302
Mohsenialhosseini SAM, Saheli M (2019) Diameter approximate best proximity pair in fuzzy normed spaces. Sahand Commun Math Anal 16(1):17–34
Sabao H, Otafudu OO (2021) Isbell convexity in fuzzy quasi-metric spaces. Khayyam J Math 7(2):266–278
Schweizer B, Sklar A (1960) Statistical metric spaces. Pacific J Math 10:313–334
Singer I (2006) Duality for nonconvex approximation and optimization. Springer, Berlin
Wang Z (2021) Fuzzy approximate m-map**s in quasi fuzzy normed spaces. Fuzzy Sets Syst 406:82–92
Wang H, Wu JR (2022) The norm of continuous linear operator between two fuzzy quasi-normed spaces. AIMS Math 7(7):11759–11771
Wang H, Wu JR, ** ZY (2023) On duality for fuzzy quasi-normed space. Comput Appl Math 42:209. https://doi.org/10.1007/s40314-023-02347-1(online)
Wu JR, Li RN (2022) Open map** and closed graph theorems in fuzzy quasi-normed spaces. IEEE Trans Fuzzy Syst 30(12):5291–5296
Zhou W, Wu JR (2022) Ekeland’s variational principle in fuzzy quasi-normed spaces. AIMS Math 7(9):15982–15991
Acknowledgements
The authors acknowledge the supports of the National Natural Science Foundation of China (Grant No. 11971343, 12071225) and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 22KJD110005).
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Communicated by Junsheng Qiao.
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Liu, H., **, Z. & Wu, J. The separation of convex sets and the Krein–Milman theorem in fuzzy quasi-normed space. Comp. Appl. Math. 43, 91 (2024). https://doi.org/10.1007/s40314-024-02593-x
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DOI: https://doi.org/10.1007/s40314-024-02593-x