Skip to main content
SpringerLink
Log in

Ternary Derivations, Stability and Physical Aspects

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

Ternary algebras and modules are vector spaces on which products of three factors are defined. In this paper, we present several physical applications of ternary structures. Some recent results on the stability of ternary derivations are reviewed. Using a fixed point method, we also establish the generalized Hyers–Ulam–Rassias stability of ternary derivations from a normed ternary algebra into a Banach tri-module associated to the generalized Jensen functional equations and prove a superstability result.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abramov, V., Kerner, R., Le Roy, B.: Hypersymmetry: A Z 3-graded generalization of supersymmetry. J. Math. Phys. 38(3), 1650–1669 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  2. Amyari, M., Moslehian, M.S.: Approximately ternary semigroup homomorphisms. Lett. Math. Phys. 77, 1–9 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Aoki, T.: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 2, 64–66 (1950)

    Article  MATH  MathSciNet  Google Scholar 

  4. Baak, C., Moslehian, M.S.: Stability of J *-homomorphisms. Nonlinear Anal. TMA 63, 42–48 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Baak, C., Moslehian, M.S.: On the stability of θ-derivations on JB *-triples. Bull. Braz. Math. Soc. 38(1), 115–127 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bars, I., Günaydin, M.: Construction of Lie algebras and Lie superalgebras from ternary algebras. J. Math. Phys. 20(9), 1977–1993 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bazunova, N., Borowiec, A., Kerner, R.: Universal differential calculus on ternary algebras. Lett. Math. Phys. 67(3), 195–206 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  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)

  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

    MathSciNet  Google Scholar 

  11. Cayley, A.: Cambr. Math. J. 4, 195–206 (1845)

    Google Scholar 

  12. Czerwik, S.: Functional Equations and Inequalities in Several Variables. World Scientific, New Jersey (2002)

    MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  14. Gajda, Z.: On stability of additive mappings. Int. J. Math. Math. Sci. 14, 431–434 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  15. Găvruta, P.: A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431–436 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hyers, D.H.: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. USA 27, 222–224 (1941)

    Article  MATH  MathSciNet  Google Scholar 

  17. Hyers, D.H., Isac, G., Rassias, Th.M.: Stability of Functional Equations in Several Variables. Birkhäuser, Basel (1998)

    MATH  Google Scholar 

  18. Jung, S.-M.: Hyers–Ulam–Rassias stability of Jensen’s equation and its application. Proc. Am. Math. Soc. 126, 3137–3143 (1998)

    Article  MATH  Google Scholar 

  19. Jung, S.-M.: Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor (2001)

    MATH  Google Scholar 

  20. Kerner, R.: Z 3-graded algebras and the cubic root of the supersymmetry translations. J. Math. Phys. 33(1), 403–411 (1992)

    Article  MathSciNet  Google Scholar 

  21. Kerner, R.: The cubic chessboard. Class. Quantum Gravity 14(1A), A203–A225 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  26. Moslehian, M.S.: Almost derivations on C *-ternary rings. Bull. Belg. Math. Soc. Simon Stevin 14(1), 135–142 (2007)

    MATH  MathSciNet  Google Scholar 

  27. Moslehian, M.S.: Asymptotic behavior of the extended Jensen equation. Studia Sci. Math. Hung. (to appear)

  28. Moslehian, M.S., Rassias, Th.M.: Stability of functional equations in non-Archimedean spaces. Appl. Anal. Discrete Math. 1(2), 325–334 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Moslehian, M.S., Székelyhidi, L.: Stability of ternary homomorphisms via generalized Jensen equation. Results Math. 49, 289–300 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  30. Park, C.: A generalized Jensen’s mapping and linear mappings between Banach modules. Bull. Braz. Math. Soc. (NS) 36(3), 333–362 (2005)

    Article  MATH  Google Scholar 

  31. Nambu, Y.: Generalized Hamiltonian dynamics. Phys. Rev. D 7, 2405–2412 (1976)

    Article  MathSciNet  Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

  33. Radu, V.: The fixed point alternative and the stability of functional equations. Semin. Fixed Point Theory 4, 91–96 (2003)

    MATH  MathSciNet  Google Scholar 

  34. Rassias, Th.M.: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  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

    Google Scholar 

  36. Rassias, Th.M. (ed.): Functional Equations, Inequalities and Applications. Kluwer Academic, Dordrecht (2003)

    MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  38. Skof, F.: Sull’approssimazione delle applicazioni localmente δ-additive. Atti Accad. Sci. Torino 117, 377–389 (1983)

    MATH  MathSciNet  Google Scholar 

  39. Takhtajan, L.: On foundation of the generalized Nambu mechanics. Commun. Math. Phys. 160(2), 295–315 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  40. Ulam, S.M.: Problems in Modern Mathematics. Wiley, New York (1964). Chap. VI, Science editions

    MATH  Google Scholar 

  41. Vainerman, L., Kerner, R.: On special classes of n-algebras. J. Math. Phys. 37(5), 2553–2565 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  42. Zettl, H.: A characterization of ternary rings of operators. Adv. Math. 48, 117–143 (1983)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Ferdowsi University, P.O. Box 1159, Mashhad, 91775, Iran

    Mohammad Sal Moslehian

  2. Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran, Iran

    Mohammad Sal Moslehian

Authors
  1. Mohammad Sal Moslehian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohammad Sal Moslehian.

Additional information

The author was in part supported by a grant from IPM (No. 85390031).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Moslehian, M.S. Ternary Derivations, Stability and Physical Aspects. Acta Appl Math 100, 187–199 (2008). https://doi.org/10.1007/s10440-007-9179-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-007-9179-x

Keywords

Mathematics Subject Classification (2000)

Search

Navigation