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Duality for Best Approximation in Fuzzy Quasi-normed Spaces

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Abstract

In the study of the classical approximation and optimization problems, the dual method is a powerful tool. The present paper initiates the problems of the duality for best approximation in fuzzy quasi-normed spaces. It gives the characterization of the nearest points, and extends the well-known Arzela formula for the distance from a point to a hyperplane in a normed space to the case of a fuzzy quasi-normed space. The obtained notions and results show the validity of dual method in the study of best approximation for fuzzy quasi-normed spaces, and will play an important role in the research of this domain.

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References

  1. Singer, I.: Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, Publishing House of the Academy of the Socialist Republic of Romania. Springer, New York (1970)

    Book  Google Scholar 

  2. Eldred, A.A., Veeramani, P.: Existence and convergence of best proximity points. J. Math. Anal. Appl. 323(2), 1001–1006 (2006)

    Article  MathSciNet  Google Scholar 

  3. Shams, M., Vaezpour, S.M.: Best approximation on probabilistic normed spaces. Chaos Solitons Fract. 41(4), 1661–1667 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  4. Goudarzi, M., Vaezpour, S.M.: Best simultaneous approximation in fuzzy normed spaces. Iran. J. Fuzzy Syst. 7(3), 87–96 (2010)

    MathSciNet  Google Scholar 

  5. Hemati, F., Mazahri, H.: Exist and uniqueness of p-best approximation in fuzzy normed spaces (Poster). In: 46th Annual Iranian Mathematics Conference, pp. 25–28, August 2015. Yazd University

  6. Lee, K.Y.: Approximation properties in fuzzy normed spaces. Fuzzy Sets Syst. 282, 115–130 (2016)

    Article  MathSciNet  ADS  Google Scholar 

  7. Vaezpour, S.M., Karimi, F.: t-Best approximation in fuzzy normed spaces. Iran. J. Fuzzy Syst. 5(2), 93–99 (2008)

    MathSciNet  Google Scholar 

  8. Moghaddam, M.A.: Best proximity pairs in fuzzy normed spaces. J. Appl. Sci. 12(9), 893–896 (2012)

    Article  ADS  Google Scholar 

  9. Mohsenialhosseini, S.A.M., Saheli, M.: Diameter approximate best proximity pair in fuzzy normed spaces. Sahand Commun. Math. Anal. 16(1), 17–34 (2019)

    Google Scholar 

  10. Moghaddam, M.A., Sistani, T.: On t-best coapproximation in fuzzy 2-normed spaces. Australian J. Basic Appl. Sci. 5(9), 2241–2248 (2011)

    Google Scholar 

  11. Reddy, B.S.: Some results on t-best approximation in fuzzy 2-normed linear spaces. Int. J. Pure Appl. Math. 72(2), 237–247 (2011)

    MathSciNet  Google Scholar 

  12. Mohiuddine, S.A.: Some new results on approximation in fuzzy 2-normed spaces. Math. Comput. Model. 53, 574–580 (2011)

    Article  MathSciNet  Google Scholar 

  13. Kavikumar, J., Khamis, A., Manian, N.S.: t-Best approximation in intuitionistic fuzzy normed spaces. In: Proceedings: Conference Paper in IEEE International Conference on Fuzzy Systems, Barcelona, Spain, 18–23 July, 2010

  14. Singer, I.: Duality for Nonconvex Approximation and Optimization. Springer, New York (2006)

    Book  Google Scholar 

  15. Krein, M.G., Nudelman, A.A.: The Markov moment problem and extremum problems,” Nauka, Moscow 1973 (in Russian). English translation: Amer. Math. Society, Providence (1977)

  16. Mustăţa, C.: On the extremal semi-Lipschitz functions. Rev. Anal. Numer. Theor. Approx. 31(1), 103–108 (2002)

    Article  MathSciNet  Google Scholar 

  17. Mustăţa, C.: On the uniqueness of the extension and unique best approximation in the dual of an asymmetric linear space. Rev. Anal. Numer. Theor. Approx. 32(2), 187–192 (2003)

    Article  MathSciNet  Google Scholar 

  18. Cobzas, S., Mustăţa, C.: Extension of bounded linear functionals and best approximation in spaces with asymmetric norm. Probat. J. 33(1), 191–212 (2004)

    MathSciNet  Google Scholar 

  19. Cobzas, S.: Functional Analysis in Asymmetric Normed Spaces. Springer, Basel (2013)

    Book  Google Scholar 

  20. Li, W. et al.: Best approximation in asymmetric normed linear spaces. In: International Conference on Information Science and Technology, March 26–28, 2011 Nanjing, Jiangsu, China

  21. Gil, C.A.: Quasi-metric properties of the dual cone of an asymmetric normed space. Results Math. 77(4), 178 (2022)

    Article  MathSciNet  Google Scholar 

  22. Bachir, M., Flores, G.: Index of symmetry and topological classification of asymmetric normed spaces. Rocky Mountain J. Math. 50(6), 1951–1964 (2020)

    Article  MathSciNet  Google Scholar 

  23. Blasco, X., et al.: Computing optimal distances to Pareto sets of multi-objective optimization problems in asymmetric normed lattices. Acta Appl. Math. 159, 75–93 (2019)

    Article  MathSciNet  Google Scholar 

  24. Alegre, C., Romaguera, S.: Characterizations of metrizable topological vector spaces and their asymmetric generalizations in terms of fuzzy (quasi-)norms. Fuzzy Sets Syst. 61, 2181–2192 (2010)

    Article  MathSciNet  Google Scholar 

  25. Bag, T., Samanta, S.K.: Finite dimensional fuzzy normed linear spaces. Fuzzy Math. 6(2), 687–705 (2003)

    MathSciNet  Google Scholar 

  26. Alegre, C., Romaguera, S.: On the uniform boundedness theorem in fuzzy quasi-normed spaces. Fuzzy Sets Syst. 282, 143–153 (2016)

    Article  MathSciNet  Google Scholar 

  27. Gao, R., Li, X.X., Wu, J.R.: The decomposition theorem for a fuzzy quasinorm. J. Math. 2020, Article ID 8845283, 7 (2020)

  28. Li, R.N., Wu, J.R.: Hahn-Banach type theorems and the separation of convex sets for fuzzy quasi-normed spaces. AIMS Math. 7(3), 3290–3302 (2021)

    Article  MathSciNet  Google Scholar 

  29. Wang, H., Wu, J.R.: The norm of continuous linear operator between two fuzzy quasi-normed spaces. AIMS Mathematics 7(7), 11759–11771 (2022)

    Article  MathSciNet  Google Scholar 

  30. Alegre, C., Romaguera, S.: On paratopological vector spaces. Acta Math. Hungar. 101, 237–261 (2003)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The first author was supported by the National Natural Science Foundation of China (11971343).

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Authors and Affiliations

  1. School of Mathematical Science, Suzhou University of Science and Technology, Suzhou, 215009, Jiangsu, People’s Republic of China

    Jian-Rong Wu, He Liu & Hua Duan

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  1. Jian-Rong Wu
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  2. He Liu
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Correspondence to Jian-Rong Wu.

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Wu, JR., Liu, H. & Duan, H. Duality for Best Approximation in Fuzzy Quasi-normed Spaces. Int. J. Fuzzy Syst. 26, 333–343 (2024). https://doi.org/10.1007/s40815-023-01562-6

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