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Ternary derivations, stability and physical aspects. (English) Zbl 1135.39014
The author states some various definitions of ternary structures [cf. M. S. Moslehian, Bull. Belg. Math. Soc. 14, No. 1, 135–142 (2007; Zbl 1132.39026), M. Amyari and M. S. Moslehian [Lett. Math. Phys. 77, No. 1, 1–9 (2006; Zbl 1112.39021)] and proves the generalized Hyers-Ulam-Rassias stability of ternary derivations associated with the generalized Jensen functional equation by using a fixed point method [see also M.S. Moslehian and L. Székelyhidi, Result. Math. 49, No. 3–4, 289–300 (2006; Zbl 1114.39010)]. Some examples of physical applications of ternary structures are given as well.

MSC:
39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
17A40 Ternary compositions
17A36 Automorphisms, derivations, other operators (nonassociative rings and algebras)
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[1] Abramov, V., Kerner, R., Le Roy, B.: Hypersymmetry: A Z 3-graded generalization of supersymmetry. J. Math. Phys. 38(3), 1650–1669 (1997) · Zbl 0872.58006
[2] Amyari, M., Moslehian, M.S.: Approximately ternary semigroup homomorphisms. Lett. Math. Phys. 77, 1–9 (2006) · Zbl 1112.39021
[3] Aoki, T.: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 2, 64–66 (1950) · Zbl 0040.35501
[4] Baak, C., Moslehian, M.S.: Stability of J *-homomorphisms. Nonlinear Anal. TMA 63, 42–48 (2005) · Zbl 1085.39026
[5] Baak, C., Moslehian, M.S.: On the stability of {\(\theta\)}-derivations on JB *-triples. Bull. Braz. Math. Soc. 38(1), 115–127 (2007) · Zbl 1127.39053
[6] Bars, I., Günaydin, M.: Construction of Lie algebras and Lie superalgebras from ternary algebras. J. Math. Phys. 20(9), 1977–1993 (1979) · Zbl 0412.17004
[7] Bazunova, N., Borowiec, A., Kerner, R.: Universal differential calculus on ternary algebras. Lett. Math. Phys. 67(3), 195–206 (2004) · Zbl 1062.46056
[8] Boo, D.-H., Oh, S.-Q., Park, C.-G., Park, J.-M.: Generalized Jensen’s equations in Banach modules over a C *-algebra and its unitary group. Taiwan. J. Math. 7(4), 641–655 (2003) · Zbl 1073.39019
[9] Brzozowski, J.A.: Some applications of ternary algebras. In: Automata and Formal Languages, vol. VIII. Salgótarján (1996). Publ. Math. Debr. 54, Suppl. 583–599 (1999) · Zbl 0981.06005
[10] Cădariu, L., Radu, V.: Fixed points and the stability of Jensen’s functional equation. J. Inequal. Pure Appl. Math. 4(1), 7 (2003) Article 4
[11] Cayley, A.: Cambr. Math. J. 4, 195–206 (1845)
[12] Czerwik, S.: Functional Equations and Inequalities in Several Variables. World Scientific, New Jersey (2002) · Zbl 1011.39019
[13] Faĭziev, V., Sahoo, P.K.: On the stability of Jensen’s functional equation on groups. Proc. Indian Acad. Sci. Math. Sci. 117(1), 31–48 (2007) · Zbl 1119.39023
[14] Gajda, Z.: On stability of additive mappings. Int. J. Math. Math. Sci. 14, 431–434 (1991) · Zbl 0739.39013
[15] Găvruta, P.: A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431–436 (1994) · Zbl 0818.46043
[16] Hyers, D.H.: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. USA 27, 222–224 (1941) · Zbl 0061.26403
[17] Hyers, D.H., Isac, G., Rassias, Th.M.: Stability of Functional Equations in Several Variables. Birkhäuser, Basel (1998) · Zbl 0907.39025
[18] Jung, S.-M.: Hyers–Ulam–Rassias stability of Jensen’s equation and its application. Proc. Am. Math. Soc. 126, 3137–3143 (1998) · Zbl 0909.39014
[19] Jung, S.-M.: Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor (2001) · Zbl 0980.39024
[20] Kerner, R.: Z 3-graded algebras and the cubic root of the supersymmetry translations. J. Math. Phys. 33(1), 403–411 (1992)
[21] Kerner, R.: The cubic chessboard. Class. Quantum Gravity 14(1A), A203–A225 (1997) · Zbl 0897.17002
[22] Kerner, R.: Ternary algebraic structures and their applications in physics. Preprint, Univ. P. and M. Curie, Paris, http://arxiv.org/math-ph/0011023 (2000)
[23] Kominek, Z.: On a local stability of the Jensen functional equation. Demonstr. Math. 22, 499–507 (1989) · Zbl 0702.39007
[24] Lee, Y.-H., Jun, K.-W.: A generalization of the Hyers–Ulam–Rassias stability of Jensen’s equation. J. Math. Anal. Appl. 238(1), 305–315 (1999) · Zbl 0933.39053
[25] Margolis, B., Diaz, J.B.: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull. Am. Math. Soc. 126, 305–309 (1968) · Zbl 0157.29904
[26] Moslehian, M.S.: Almost derivations on C *-ternary rings. Bull. Belg. Math. Soc. Simon Stevin 14(1), 135–142 (2007) · Zbl 1132.39026
[27] Moslehian, M.S.: Asymptotic behavior of the extended Jensen equation. Studia Sci. Math. Hung. (to appear) · Zbl 1274.39066
[28] Moslehian, M.S., Rassias, Th.M.: Stability of functional equations in non-Archimedean spaces. Appl. Anal. Discrete Math. 1(2), 325–334 (2007) · Zbl 1257.39019
[29] Moslehian, M.S., Székelyhidi, L.: Stability of ternary homomorphisms via generalized Jensen equation. Results Math. 49, 289–300 (2006) · Zbl 1114.39010
[30] Park, C.: A generalized Jensen’s mapping and linear mappings between Banach modules. Bull. Braz. Math. Soc. (NS) 36(3), 333–362 (2005) · Zbl 1093.47040
[31] Nambu, Y.: Generalized Hamiltonian dynamics. Phys. Rev. D 7, 2405–2412 (1976) · Zbl 1027.70503
[32] Okubo, S.: Triple products and Yang–Baxter equation. I & II. Octonionic and quaternionic triple systems. J. Math. Phys. 34(7), 3273–3291 (1993) and 34(7), 3292–3315 · Zbl 0790.15028
[33] Radu, V.: The fixed point alternative and the stability of functional equations. Semin. Fixed Point Theory 4, 91–96 (2003) · Zbl 1051.39031
[34] Rassias, Th.M.: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978) · Zbl 0398.47040
[35] Rassias, Th.M.: Problem 16; 2, Report of the 27th international symp. on functional equations. Aequ. Math. 39, 292–293 (1990) and 39, 309
[36] Rassias, Th.M. (ed.): Functional Equations, Inequalities and Applications. Kluwer Academic, Dordrecht (2003) · Zbl 1047.39001
[37] Rassias, Th.M., Šemrl, P.: On the behaviour of mappings which do not satisfy Hyers–Ulam stability. Proc. Am. Math. Soc. 114, 989–993 (1992) · Zbl 0761.47004
[38] Skof, F.: Sull’approssimazione delle applicazioni localmente {\(\delta\)}-additive. Atti Accad. Sci. Torino 117, 377–389 (1983) · Zbl 0794.39008
[39] Takhtajan, L.: On foundation of the generalized Nambu mechanics. Commun. Math. Phys. 160(2), 295–315 (1994) · Zbl 0808.70015
[40] Ulam, S.M.: Problems in Modern Mathematics. Wiley, New York (1964). Chap. VI, Science editions · Zbl 0137.24201
[41] Vainerman, L., Kerner, R.: On special classes of n-algebras. J. Math. Phys. 37(5), 2553–2565 (1996) · Zbl 0864.17002
[42] Zettl, H.: A characterization of ternary rings of operators. Adv. Math. 48, 117–143 (1983) · Zbl 0517.46049
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