Page 53 - Textos de Matemática Vol. 44

ON APPROXIMATE GENERALIZED DERIVATIONS 43
Proof. Using the last two equalities in Theorem 3.1, we see that d = D and g = G.
Since every Banach algebra with an approximate identity, as a Banach bi- module over itself, is without order, we have the next result.
Corollary 4.6. Let A be a Banach algebra with a bounded approximate identity and ε1 , ε2 , ν1 , ν2 nonnegative real numbers. Assume that d, g : A → A satisfy
∥d(a+b)−d(a)−d(b)∥≤ε1, ∥g(a+b)−g(a)−g(b)∥≤ε2, ∥d(ab) − d(a)b − ad(b)∥ ≤ ν1, ∥g(ab) − g(a)b − ad(b)∥ ≤ ν2
and
for all a, b ∈ A. Then d is a derivation on A and g is a generalized derivation
on A with an associate derivation d. Acknowledgments
The second author is financially supported by the Tusi Mathematical Research Group (TMRG), Mashhad, Iran.
References
[1] M. Amyari, C. Baak and M. S. Moslehian, Nearly ternary derivations, Taiwanese J. Math. 11 (2007), 1417–1424.
[2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan. 2 (1950), 64–66.
[3] R. Badora, On approximate ring homomorphisms, J. Math. Anal. Appl. 276 (2002), 589–597.
[4] , On approximate derivations, Math. Inequal. Appl. 9 (2006), 167–173.
[5] J. Baker, J. Lawrence and F. Zorzitto, The stability of the equation f(x+y) = f(x)f(y),
Proc. Amer. Math. Soc. 74 (1979), 242–246.
[6] D. G. Bourgin, Approximately isometric and multiplicative transformations on contin-
uous function rings, Duke Math. J. 16 (1949), 385–397.
[7] M. Breˇsar, On the distance of the compositum of two derivations to the generalized
derivations, Glasgow Math. J. 33 (1991) 89–93.
[8] S. Czerwik, Functional equations and inequalities in several variables, World Scientific
Publishing, New Jersey, 2002.
[9] H. G. Dales Banach Algebras and Automatic Continuity, Clarendon Press, Oxford, 2000.
[10] A. Foˇsner, On the generalized Hyers-Ulam stability of module left (m,n)-derivations, Aequationes Math. 84 (2012), no. 1-2, 91–98.
[11] M. E. Gordji and M. S. Moslehian, A trick for investigation of approximate derivations, Math. Commun. 15 (2010), 99–105.
[12] B. Hvala, Generalized derivations in rings, Comm. Algebra 26 (1998) 1147–1166.
[13] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci.
U.S.A. 27 (1941), 222–224.