Motivated by the fact that the fuzzy quasi-normed space provides a suitable framework for complexity analysis and has important roles in discussing some questions in theoretical computer science, this paper aims to study the nearest point problems in fuzzy quasi-normed spaces. First, by using the theory of dual space and the separation theorem of convex sets, the properties of the fuzzy distance from a point to a set in a fuzzy quasi-normed space are studied comprehensively. Second, more properties of the nearest point are given, and the existence, uniqueness, characterizations, and qualitative properties of the nearest points are obtained. The results obtained in this paper are of great significance for expanding the application fields of optimization theory.
Citation: Jian-Rong Wu, He Liu. The nearest point problems in fuzzy quasi-normed spaces[J]. AIMS Mathematics, 2024, 9(3): 7610-7626. doi: 10.3934/math.2024369
Motivated by the fact that the fuzzy quasi-normed space provides a suitable framework for complexity analysis and has important roles in discussing some questions in theoretical computer science, this paper aims to study the nearest point problems in fuzzy quasi-normed spaces. First, by using the theory of dual space and the separation theorem of convex sets, the properties of the fuzzy distance from a point to a set in a fuzzy quasi-normed space are studied comprehensively. Second, more properties of the nearest point are given, and the existence, uniqueness, characterizations, and qualitative properties of the nearest points are obtained. The results obtained in this paper are of great significance for expanding the application fields of optimization theory.
| [1] |
C. Alegre, S. Romaguera, On paratopological vector spaces, Acta Math. Hungar., 101 (2003), 237–261. https://doi.org/10.1023/B:AMHU.0000003908.28255.22 doi: 10.1023/B:AMHU.0000003908.28255.22
|
| [2] |
C. Alegre, S. Romaguera, Characterizations of metrizable topological vector spaces and their asymmetric generalizations in terms of fuzzy (quasi-)norms, Fuzzy Sets Syst., 161 (2010), 2181–2192. https://doi.org/10.1016/j.fss.2010.04.002 doi: 10.1016/j.fss.2010.04.002
|
| [3] |
H. H. Bauschke, H. Ouyang, X. Wang, Finding best approximation pairs for two intersections of closed convex sets, Comput. Optim. Appl., 81 (2022), 289–308. https://doi.org/10.1007/s10589-021-00324-0 doi: 10.1007/s10589-021-00324-0
|
| [4] |
H. H. Bauschke, H. Ouyang, X. Wang, Best approximation mappings in Hilbert spaces, Math. Program., 195 (2022), 855–901. https://doi.org/10.1007/s10107-021-01718-y doi: 10.1007/s10107-021-01718-y
|
| [5] |
J. M. Borwein, S. Fitzpatrick, Existence of nearest points in Banach spaces, Can. J. Math., 41 (1989), 702–720. https://doi.org/10.4153/CJM-1989-032-7 doi: 10.4153/CJM-1989-032-7
|
| [6] | S. Cobzas, Functional analysis in asymmetric normed spaces, Basel: Springer, 2013 |
| [7] |
S. Cobzas, C. Mustăţa, Extension of bounded linear functionals and best approximation in spaces with asymmetric norm, Rev. Anal. Numer. Theor. Approx., 33 (2004), 39–50. https://doi.org/10.33993/jnaat331-757 doi: 10.33993/jnaat331-757
|
| [8] |
A. A. Eldred, P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl., 323 (2006), 1001–1006. https://doi.org/10.1016/j.jmaa.2005.10.081 doi: 10.1016/j.jmaa.2005.10.081
|
| [9] |
R. Gao, X. X. Li, J. R. Wu, The decomposition theorem for a fuzzy quasi norm, J. Math., 2020 (2020), 8845283. https://doi.org/10.1155/2020/8845283 doi: 10.1155/2020/8845283
|
| [10] | M. Goudarzi, S. M. Vaezpour, Best simultaneous approximation in fuzzy normed spaces, Iran. J. Fuzzy Syst., 7 (2010), 87–96. |
| [11] | M. G. Krein, A. A. Nudelman, The Markov moment problem and extremal problems: ideas and problems of P. L. Čebyšev and A. A. Markov and their further development, American Mathematical Society, 1977. |
| [12] |
R. N. Li, J. R. Wu, Hahn-Banach type theorems and the separation of convex sets for fuzzy quasi-normed spaces, AIMS Math., 7 (2022), 3290–3302. https://doi.org/10.3934/math.2022183 doi: 10.3934/math.2022183
|
| [13] |
H. Liu, Z. Y. Jin, J. R. Wu, The separation of convex sets and the Krein-Milman theorem in fuzzy quasi-normed space, Comput. Appl. Math., 43 (2024), 2024. https://doi.org/10.1007/s40314-024-02593-x doi: 10.1007/s40314-024-02593-x
|
| [14] |
S. A. Mohiuddine, Some new results on approximation in fuzzy 2-normed spaces, Math. Comput. Model., 53 (2011), 574–580. https://doi.org/10.1016/j.mcm.2010.09.006 doi: 10.1016/j.mcm.2010.09.006
|
| [15] |
C. Mustăţa, On the extremal semi-Lipschitz functions, Rev. Anal. Numer. Theor. Approx., 31 (2002), 103–108. https://doi.org/10.33993/jnaat311-712 doi: 10.33993/jnaat311-712
|
| [16] |
C. Mustăţa. On the uniqueness of the extension and unique best approximation in the dual of an asymmetric linear space, Rev. Anal. Numer. Theor. Approx., 32 (2003), 187–192. https://doi.org/10.33993/jnaat322-747 doi: 10.33993/jnaat322-747
|
| [17] | B. Schweizer, A. Sklar, Statistical metric spaces, Pacific. J. Math., 10 (1960), 314–334. |
| [18] |
M. Shams, S. M. Vaezpour, Best approximation on probabilistic normed spaces, Chaos Soliton. Fract., 41 (2009), 1661–1667. https://doi.org/10.1016/j.chaos.2008.07.009 doi: 10.1016/j.chaos.2008.07.009
|
| [19] | I. Singer, Best approximation in normed linear spaces by elements of linear subspaces, Berlin: Springer, 1970. https://doi.org/10.1007/978-3-662-41583-2 |
| [20] | I. Singer, Duality for nonconvex approximation and optimization, Springer Science & Business Media, 2007. |
| [21] |
J. R. Wu, H. Liu, H. Duan, Duality for best approximation in fuzzy quasi-normed spaces, Int. J. Fuzzy Syst., 2023. https://doi.org/10.1007/s40815-023-01562-6 doi: 10.1007/s40815-023-01562-6
|
Jian-Rong Wu, He Liu. The nearest point problems in fuzzy quasi-normed spaces[J]. AIMS Mathematics, 2024, 9(3): 7610-7626. doi: 10.3934/math.2024369
DownLoad: