We introduce the definition of non-Archimedean 2-fuzzy 2-normed spaces and the concept of isometry which is appropriate to represent the notion of area preserving mapping in the spaces above. And then we can get isometry when a mapping satisfies AOPP and (*) (in article) by applying the Benz’s theorem about the Aleksandrov problem in non-Archimedean 2-fuzzy 2-normed spaces.
References
[1]
Benz, W. and Berens, H. (1987) A Contribution to a Theorem of Ulam and Mazur. Aequationes Mathematicae, 34, 61-63. https://doi.org/10.1007/BF01840123
[2]
Bag, T. and Samanta, S.K. (2003) Finite Dimensional Fuzzy Normed Linear Spaces. The Journal of Fuzzy Mathematics, 11, 687-705.
[3]
Chu, H.Y., Park, C.G. and Park, W.G. (2004) The Aleksandrow Problem in Linear 2-Normed Spaces. Journal of Mathematical Analysis and Applications, 289, 666-672.
https://doi.org/10.1016/j.jmaa.2003.09.009
[4]
Somasundaram, R.M. and Beaula, T. (2009) Some Aspects of 2-Fuzzy 2-Normed Linear Spaces. Bulletin of the Malaysian Mathematical Sciences Society, 32, 211-221.
[5]
Zheng, F.H. and Ren, W.Y. (2014) The Aleksandrow Problem in Quasi Convex Normed Linear Space. Acta Scientiarum Natueralium University Nankaiensis, No. 3, 49-56.
[6]
Huang, X.J. and Tan, D.N. (2017) Mapping of Conservative Distances in p-Normed Spaces ( ). Bulletin of the Australian Mathematical Society, 95, 291-298.
https://doi.org/10.1017/S0004972716000927
[7]
Ma, Y.M. (2000) The Aleksandrov Problem for Unit Distance Preserving Mapping. Acta Mathematica Science, 20B, 359-364.
https://doi.org/10.1016/S0252-9602(17)30642-2
[8]
Wang, D.P., Liu, Y.B. and Song, M.M. (2012) The Aleksandrov Problem on Non-Archimedean Normed Spaces. Arab Journal of Mathematical Science, 18, 135-140. https://doi.org/10.1016/j.ajmsc.2011.10.002
[9]
Ma, Y.M. (2016) The Aleksandrov-Benz-Rassias Problem on Linear n-Normed Spaces. Monatshefte für Mathematik, 180, 305-316.
https://doi.org/10.1007/s00605-015-0786-8
[10]
Huang, X.J. and Tan, D.N. (2018) Mappings of Preserves n-Distance One in N-Normed Spaces. Aequations Mathematica, 92, 401-413.
https://doi.org/10.1007/s00010-018-0539-6
[11]
Xu, T.Z. (2013) On the Mazur-Ulanm Theorem in Non-Archimedean Fuzzy n-Normed Spaces. ISRN Mathematical Analysis, 67, 1-7.
https://doi.org/10.1155/2013/814067
[12]
Chang, L.F. and Song, M.M. (2014) On the Mazur-Ulam Theorem in Non-Archimedean Fuzzy 2-Normed Spaces. Mathematica Applicata, 27, 355-359.
[13]
Alaca (2010) New Perspective to the Mazur-Ulam Problem in 2-Fuzzy 2-Normed Linear Spaces. Iranian Journal of Fuzzy Systems, 7, 109-119.
[14]
Park, C. and Alaca, C. (2013) Mazur-Ulam Theorem under Weaker Conditions in the Framework of 2-Fuzzy 2-Normed Linear Spaces. Journal of Inequalities and Applications, 2018, 78. https://doi.org/10.1186/1029-242X-2013-78
[15]
Hensel, K. (1897) über eine neue Begründung der Theorie der algebraischen Zahlen. Jahresbericht der Deutschen Mathematiker-Vereinigung, 6, 83-88.
