×

Mappings preserving unit distance on Heisenberg group. (English) Zbl 1321.39035

Summary: Let \(H^{m}\) be a Heisenberg group provided with a norm \(\rho\). A mapping \(f : H^{m} \to H^{m}\) is called preserving the distance \(n\) if for all \(x, y\) of \(H^{m}\) with \(\rho(x^{-1}y) = n\) then \(\rho(f(x)^{-1}f(y)) = n\). We obtain some results for the Aleksandrov problem in the Heisenberg group.

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A.D. Aleksandrov, Mappings of families of sets, Soviet Math. 11 (1970), 116-120.; · Zbl 0213.48903
[2] M. Bavand Savadkouhi, M. Eshaghi Gordji, J. M. Rassias, and N. Ghobadipour, Approximate ternary Jordan derivations on Banach ternary algebras, Journal of Mathematical Physics 50, 042303, (2009).; · Zbl 1214.46034
[3] R.L. Bishop, Characterizing motions by unit distance invariance, Math. Magazine 46 (1973), 148-151.; · Zbl 0262.50001
[4] A. Bodaghi, Cubic derivations on Banach algebras, Acta Math. Vietnam., No 4, Vol.38 (2013), 517{528.;} · Zbl 1307.39015
[5] A. Charifi, Iz. El-Fassi, B. Bouikhalene, S. Kabbaj, On the approximate solutions of the Pex- iderized Golab-Schinzel functional equation, Acta Universitatis Apulensis, No. 38/2014, pp. 55{66.;} · Zbl 1340.39037
[6] Ab. Chahbi, A. Charifi, B. Bouikhalene, S. Kabbaj, Operatorial approach to the non- Archimedean stability of a Pexider K-quadratic functional equation, Arab Journal of Math- ematical Sciences, Available online 17 January 2014, doi:10.1016/j.ajmsc.2014.01.001.; · Zbl 1308.39022
[7] X.Y. Chen, M.M. Song, Characterizations on isometries in linear n-normed spaces, Nonlinear Anal 2010;72:1895-901.; · Zbl 1196.46011
[8] H.Y. Chu, C.G. Park, W.G. Park, The Aleksandrov problem in linear 2-normed spaces, J. Math. Anal. Appl. 289 (2004) 666{672.;} · Zbl 1045.46002
[9] G.G. Ding, On isometric extensions and distance one preserving mappings, Taiwanese J. Math. 10 (1) (2006), 243{249.;} · Zbl 1107.46008
[10] D. V. Isangulova , The class of mappings with bounded specific oscillation, and integrability of mappings with bounded distortion on Carnot groups, Sibirsk. Mat. Zh. 48 (2007), no. 2, 313{334; English translation in: Siberian Math. J. 48 (2007), no. 2, 249-267.;} · Zbl 1164.30362
[11] D. V. Isangulova and S. K. Vodopyanov, Sharp geometric rigidity of isometries on Heisenberg groups, Mathematische Annalen (2013), 1{29.;} · Zbl 1279.53030
[12] A.V. Kuzminykh, Mappings preserving the distance 1, Siberian Math. J. 20 (1979), 417421.;
[13] Y.M. Ma, The Aleksandrov problem for unit distance preserving mappings, Acta Math. Sci. 20 (3) (2000), 359{364.;} · Zbl 0973.46011
[14] Yumei Ma, Isometry on linear n-normed spaces, Annales Academi Scientiarum Fennic Math- ematica, Vol. 39 (2014), 973{981.;} · Zbl 1310.46013
[15] Y.M. Ma, J.Y. Wang, On the A.D. Aleksandrov problem of isometric mapping, J. Math. Res. Exposition 23 (4) (2003), 623{630.;} · Zbl 1159.46303
[16] B. Mielnik, Th.M. Rassias, On the Aleksandrov problem of conservative distances, Proc. Amer. Math. Soc. 116 (1992), 1115{1118.;} · Zbl 0769.51005
[17] J.M. Rassias, S. Xiang, M.J. Rassias, On the Aleksandrov and triangle isometry Ulam stability problem, Int. J. Appl. Math. Stat. 7 (2007), 133{142.;}
[18] Th.M. Rassias, P. Semrl, On the Mazur-Ulam problem and the Aleksandrov problem for unit distance preserving mappings, Proc. Amer. Math. Soc. 118 (1993), 919{925.;} · Zbl 0780.51010
[19] K. Ravi, E. Thandapani, B. V. Senthil Kumar, Solution and stability of a reciprocal type functional equation in several variables, J. Nonlinear Sc. Appl., 7 (2014), 18{27.;} · Zbl 1296.39027
[20] K. Ravi, M. Arunkumar, P. Narasimman, Fuzzy stability of an additive functional equation, Int. Journ. Math. and Stat., Vol. 9, No A11 (2011), 88{105.;}
[21] G. Zamani Eskandani, Pasc Gavruta, John M. Rassias and Ramazan Zarghami, Generalized Hyers-Ulam stability for a general mixed functional equation in quasi-normed spaces, Mediterr. J. Math. 8 (2011), 331-348.; · Zbl 1236.39026
[22] Tian Zhou Xu, John Michael Rassias, and Wan Xin Xu, Intuitionistic fuzzy stability of a general mixed additive-cubic equation, Journal of Mathematical Physics 51, 063519, (2010).; · Zbl 1311.46066
[23] D. Wang, Y. Liu and M. Song, The Aleksandrov problem on non-Archimedean normed space, Arab Journal of Mathematical Sciences, 18(2) (2012), 135{140.;} · Zbl 1254.46072
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.