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ADDITIVE FUNCTIONAL INEQUALITIES AND DERIVATIONS ON HILBERT C*-MODULES

Published online by Cambridge University Press:  05 March 2013

FRIDOUN MORADLOU
FRIDOUN MORADLOU*
Affiliation:
Department of Mathematics, Sahand University of Technology, Tabriz, Iran e-mail: [email protected]
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Abstract

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In this paper we investigate the following functional inequality

$ \begin{eqnarray*} \| f(x-y-z) - f(x-y+z) + f(y) +f(z)\| \leq \|f(x+y-z) - f(x)\| \end{eqnarray*}$
in Banach spaces, and employing the above inequality we prove the generalized Hyers–Ulam stability of derivations in Hilbert C*-modules.

Keywords

Primary 39B7246L0846B0313N1547Jxx
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 55 , Issue 2 , May 2013 , pp. 341 - 348
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013

References

REFERENCES

1.Aczél, J. and Dhombres, J., Functional equations in several variables (Cambridge University Press, Cambridge, UK, 1989).Google Scholar
2.Amyari, M. and Moslehian, M. S., Hyers–Ulam–Rassias stability of derivations on Hilbert C*-modules, Contemp. Math. 427 (2007), 31–39.Google Scholar
3.Aoki, T., On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66.Google Scholar
4.Bourgin, D. G., Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57 (1951), 223–237.CrossRefGoogle Scholar
5.Czerwik, S., The stability of the quadratic functional equation, in Stability of mappings of Hyers–Ulam type (Rassias, Th. M. and Tabor, J., Editors) (Hadronic Press, Palm Harbor, FL, 1994), pp. 81–91.Google Scholar
6.Fechner, W., Stability of a functional inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 71 (2006), 149–161.CrossRefGoogle Scholar
7.Gǎvruta, P., A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436.CrossRefGoogle Scholar
8.Ghaemi, M. B. and Alizadeh, B., Superstability of higher derivations on Hilbert C*-modules, Int. J. Nonlinear Anal. Appl. 1 (2) (2010), 36–43.Google Scholar
9.Gilányi, A., Eine zur Parallelogrammgleichung äquivalente Ungleichung, Aequationes Math. 62 (2001), 303–309.Google Scholar
10.Gilányi, A., On a problem by K. Nikodem, Math. Inequal. Appl. 5 (2002), 707–710.Google Scholar
11.Hyers, D. H., On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA 27 (1941), 222–224.CrossRefGoogle ScholarPubMed
12.Kadison, R. V. and Pedersen, G., Means and convex combinations of unitary operators, Math. Scand. 57 (1985), 249–266.Google Scholar
13.Kaplansky, I., Modules over operator algebras (American Mathematical Society Providence, RI, 1997).Google Scholar
14.Kim, H.-M. and Rassias, J. M., Generalization of Ulam stability problem for Euler–Lagrange quadratic mappings, J. Math. Anal. Appl. 336 (2007), 277–296.Google Scholar
15.Maligranda, L., A result of Tosio Aoki about a generalization of Hyers–Ulam stability of additive functions – a question of priority, Aequationes Math. 75 (2008) 289–296.Google Scholar
16.Moradlou, F., Najati, A. and Vaezi, H., Stability of homomorphisms and derivations on C*-Ternary rings associated to an Euler–Lagrange type additive mapping, Result. Math. 55 (2009), 469–486.CrossRefGoogle Scholar
17.Moradlou, F., Vaezi, H. and Eskandani, G. Z., Hyers–Ulam–Rassias stability of a quadratic and additive functional equation in quasi-Banach spaces, Mediterr. J. Math. 6 (2) (2009), 233–248.Google Scholar
18.Moradlou, F., Vaezi, H. and Park, C., Fixed points and stability of an additive functional equation of n-Apollonius type in C*-algebras, Abstr. Appl. Anal. 2008 (2008), Article ID 672618, 13 p. doi:10.1155/2008/672618.CrossRefGoogle Scholar
19.Park, C., On an approximate automorphism on a C*-algebra, Proc. Amer. Math. Soc. 132 (2004), 1739–1745.Google Scholar
20.Park, C., Homomorphisms between Poisson JC*-algebras, Bull. Braz. Math. Soc. 36 (2005), 79–97.Google Scholar
21.Park, C. and Rassias, Th. M., Hyers–Ulam stability of a generalized Apollonius-type quadratic mapping, J. Math. Anal. Appl. 322 (2006), 371–381.Google Scholar
22.Park, C. and Rassias, Th. M., Homomorphisms in C*-ternary algebras and JB*-triples, J. Math. Anal. Appl. 337 (2008), 13–20.Google Scholar
23.Rassias, J. M., On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (1982), 126–130.Google Scholar
24.Rassias, J. M., On approximation of approximately linear mappings by linear mappings, Bull. Sci. Math. 108 (1984), 445–446.Google Scholar
25.Rassias, J. M., Solution of a problem of Ulam, J. Approx. Theory 57 (1989), 268–273.CrossRefGoogle Scholar
26.Rassias, J. M. and Rassias, M. J., Refined Ulam stability for Euler–Lagrange type mappings in Hilbert spaces, Int. J. Appl. Math. Stat. 7 (2007), 126–132.Google Scholar
27.Rassias, Th. M., On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300.CrossRefGoogle Scholar
28.Rassias, Th. M., The problem of S. M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl. 246 (2000), 352–378.Google Scholar
29.Rassias, Th. M., On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), 264–284.Google Scholar
30.Rätz, J., On inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 66 (2003), 191–200.CrossRefGoogle Scholar
31.Ulam, S. M., A collection of the mathematical problems (Interscience, New York, 1960).Google Scholar