Glasnik Matematicki, Vol. 50, No. 1 (2015), 77-99.

ON APPROXIMATE GENERALIZED LIE DERIVATIONS

Janusz Brzdęk and Ajda Fošner

Department of Mathematics, Pedagogical University, Podchorążych 2, 30-084 Kraków, Poland
e-mail: jbrzdek@up.krakow.pl

Faculty of Management, University of Primorska, Cankarjeva 5, SI-6104 Koper, Slovenia
e-mail: ajda.fosner@fm-kp.si


Abstract.   Motivated by the notion of the Hyers-Ulam stability, we prove results that are efficient tools for the study of approximate generalized Lie derivations on Lie algebras. We also provide simple examples of applications of our outcomes. In particular, we obtain some auxiliary results on the stability of the additive Cauchy equation.

2010 Mathematics Subject Classification.   16W20, 16W25, 39B62, 39B82.

Key words and phrases.   Stability, normed algebra, Banach bimodule, Lie derivation, generalized Lie derivation.


Full text (PDF) (access from subscribing institutions only)

DOI: 10.3336/gm.50.1.07


References:

  1. M. Amyari, C. Baak and M. S. Moslehian, Nearly ternary derivations, Taiwanese J. Math. 11 (2007), 1417-1424.
    MathSciNet    

  2. T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64-66.
    MathSciNet     CrossRef

  3. R. Badora, On approximate derivations, Math. Inequal. Appl. 9 (2006), 167-173.
    MathSciNet     CrossRef

  4. Z. Boros and E. Gselmann, Hyers-Ulam stability of derivations and linear functions, Aequationes Math. 80 (2010), 13-25.
    MathSciNet     CrossRef

  5. N. Brillouët-Belluot, J. Brzdęk and K. Ciepliński, On some recent developments in Ulam's type stability, Abstr. Appl. Anal. 2012 (2012), Article ID 716936, 41 pages.
    MathSciNet    

  6. J. Brzdęk, Remarks on hyperstability of the Cauchy functional equation, Aequationes Math., DOI 10.1007/s00010-012-0168-4.
    MathSciNet     CrossRef

  7. J. Brzdęk, J. Chudziak and Z. Páles, A fixed point approach to stability of functional equations, Nonlinear Anal. 74 (2011), 6728-6732.
    MathSciNet     CrossRef

  8. J. Brzdęk and A. Fošner, Remarks on the stability of Lie homomorphisms, J. Math. Anal. Appl. 400 (2013), 585-596. CrossRef

  9. P. Fischer and Z. Słodkowski, Christensen zero sets and measurable convex functions, Proc. Amer. Math. Soc. 79 (1980), 449-453.
    MathSciNet     CrossRef

  10. M. E. Gordji and M. S. Moslehian, A trick for investigation of approximate derivations, Math. Commun. 15 (2010), 99-105.
    MathSciNet    

  11. S. Hejazian, H. Mahdavian Rad and M. Mirzavaziri, (δ,ε)-double derivations on Banach algebras, Ann. Funct. Anal. 1 (2010), 103-111.
    MathSciNet     CrossRef

  12. B. Hvala, Generalized Lie derivations on prime rings, Taiwan. J. Math 11 (2007), 1425-1430.

  13. D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222-224.
    MathSciNet     CrossRef

  14. D. H. Hyers, G. Isac and Th. M. Rassias, Stability of functional equations in several variables, Birkhäuser, Boston, 1998.
    MathSciNet    

  15. W. Jabłoński, On a class of sets connected with a convex function, Abh. Math. Sem. Univ. Hamburg 69 (1999), 205-210.
    MathSciNet     CrossRef

  16. W. Jabłoński, Sum of graphs of continuous functions and boundedness of additive operators, J. Math. Anal. Appl. 312 (2005), 527-534.
    MathSciNet     CrossRef

  17. S.-M. Jung, Hyers-Ulam-Rassias stability of functional equations in nonlinear analysis, Springer Optimization and Its Applications, 48. Springer, New York, 2011.
    MathSciNet    

  18. K.-W. Jun and D.-W. Park, Almost derivations on the Banach algebra Cn[0,1], Bull. Korean Math. Soc. 33 (1996), 359-366.
    MathSciNet    

  19. M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities: Cauchy’s Equation and Jensen’s Inequality (Second Edition), Birkhäuser, Boston, 2009.
    MathSciNet    

  20. M. Mirzavaziri and M. S. Moslehian, Automatic continuity of σ-derivations on C*-algebras, Proc. Amer. Math. Soc. 134 (2006), 3319-3327.
    MathSciNet     CrossRef

  21. T. Miura, H. Oka, G. Hirasawa and S.-E. Takahasi, Superstability of multipliers and ring derivations on Banach algebras, Banach J. Math. Anal. 1 (2007), 125-130.
    MathSciNet     CrossRef

  22. M. S. Moslehian, Ternary derivations, stability and physical aspects, Acta Appl. Math. 100 (2008), 187-199.
    MathSciNet     CrossRef

  23. A. Nakajima, On generalized higher derivations, Turkish J. Math. 24 (2000), 295-311.
    MathSciNet    

  24. C. Park, Linear derivations on Banach algebras, Nonlinear Funct. Anal. Appl. 9 (2004), 359–-368.
    MathSciNet    

  25. Th. M. Rassias,On the stability of the linear mappings in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.
    MathSciNet     CrossRef

  26. S. M. Ulam, Problems in modern mathematics, John Wiley & Sons, New York, 1964.
    MathSciNet    

Glasnik Matematicki Home Page