No CrossRef data available.
Article contents
Isometry on Linear n-G-quasi Normed Spaces
Published online by Cambridge University Press: 20 November 2018
Abstract
Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
This paper generalizes the Aleksandrov problem: the Mazur-Ulam theorem on 
$n-G$-quasi normed spaces. It proves that a one-
$n$-distance preserving mapping is an $n$-isometry if and only if it has the zero-$n-G$-quasi preserving property, and two kinds of $n$-isometries on $n-G$-quasi normed space are equivalent; we generalize the Benz theorem to $n$-normed spaces with no restrictions on the dimension of spaces.
- Type
- Research Article
- Information
- Copyright
- Copyright © Canadian Mathematical Society 2017
References
[1]
Aleksandrov, A. D., Mappings of families of sets.
Dokl. Akad. Nauk SSSR
191(1970), 503–506. Google Scholar
[2]
Benz, W., A contribution to a theorem ofUlam and Mazur.
Aequationes Math.
34(1987), 61–63. http://dx.doi.org/1 0.1007/BF01 840123
Google Scholar
[3]
Chu, H., Choi, S., and Kang, D., Mapping of conservative distances in linear n-normed spaces.
Nonlinear Anal.
70(2009), no. 3,1168-1174. http://dx.doi.org/10.1016/j.na.2OO8.O2.002
Google Scholar
[4]
Chu, H., Lee, K., and Park, C., On the Aleksandrov problem in linear n-normed spaces.
Nonlinear Anal.
59(2004), no. 7,1001-1011. Google Scholar
[5]
Ekariani, S., Gunawan, H., and Idris, M., A contractive mapping theorem on the n-normed space of p-summable sequences.
J. Math. Anal.
4(2013) 1–7. Google Scholar
[6]
Gao, J., On the Aleksandrov problem of distance preserving mapping.
J. Math. Anal. Appl.
352(2009), no. 2, 583–590. http://dx.doi.org/10.1016/j.jmaa.2008.10.022
Google Scholar
[7]
Gehér, G. P., On n-norm preservers and the Aleksandrov conservative n-distance problem. arxiv:1 507.05046 Google Scholar
[8]
Jing, Y., The Aleksandrov problem in p-normed spaces (0 <p < 1). Acta Sci. Nat. Univ. Nankai.
4(2008), 91–96. Google Scholar
[9]
Mazur, S. and Ulam, S., Sur les transformationes isométriques d'espaces vectoriels normes.
C. R. Acad. Sci. Paris.
194(1932), 946–948. Google Scholar
[10]
Ma, Y., The Aleksandrov problem for unit distance preserving mapping.
Acta Math. Sci. Ser. B Engl. Ed.
20(2000), no. 3, 359–364.Google Scholar
[11]
Ma, Y., The Aleksandrov problem and The Mazur- Ulam theorem on linear n-normed space.
Bull. Korean Math. Soc.
50(2013), no. 5, 1631–1637. http://dx.doi.org/10.4134/BKMS.2013.50.5.1631
Google Scholar
[12]
Ma, Y., Isometry on linear n-normed spaces.
Ann. Acad. Sci. Fenn. Math.
39(2014),no. 2, 973–981. http://dx.doi.org/10.5186/aasfm.2014.3941
Google Scholar
[13]
Ma, Y., The Aleksandrov-Benz-Rassias problem on linear n-normed spaces,
Monatsh. Math.
180(2016), 305–316. http://dx.doi.org/!0.1007/s00605-015-0786-8
Google Scholar
[14]
Park, C. and Alaca, C., Anew version ofMazur- Ulam theorem under weaker conditions in linear n-normed spaces.
J. Comput. Anal. Appl.
16(2014), no. 5, 827–832. Google Scholar
[15]
Park, C. and Rassias, T. M., Isometries on linear n-normed spaces.
JIPAM. J. Inequal. Pure Appl. Math.
7(2006), Article 168, 7 pp. (electronic).Google Scholar
[16]
Park, C. and Rassias, T. M., Isometric additive in generalized quasi-Banach spaces.
Banach J. Math. Anal.
2(2008), no. 1, 59–69. http://dx.doi.org/10.15352/bjma/1240336274
Google Scholar
[17]
Rassias, T. M. and Semrl, P., On the Mazur-Ulam theorem and the Aleksandrov problem for unit distance preserving mappings.
Proc. Amer. Math. Soc.
132(1993), no. 3, 919–925. http://dx.doi.org/10.1090/S0002-9939-1993-1111437-6
Google Scholar
[18]
Yunchen, X. and Meimei, S., Characterizations on isometries in linear n-normed spaces.
Nonlinear Anal.
72(2010), 1895–1901. http://dx.doi.org/10.1016/j.na.2009.09.029
Google Scholar
[19]
Zheng, F. and Ren, W.
The Aleksandrov problem in quasi convex normed linear space.
Acta Sci. Natur. Nankai Univ.
47(2014), no. 3, 49–56.Google Scholar
You have
Access