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Approximate linear derivations and functional inequalities with applications. (English) Zbl 1252.39029

The authors consider approximate derivations and prove the interesting result that any approximate linear derivation on a semisimple Banach algebra is continuous (it is known that any linear derivation on a semisimple Banach algebra is continuous).
Also some results concerning the stability of special functional equations are presented.

MSC:

39B62 Functional inequalities, including subadditivity, convexity, etc.
39B82 Stability, separation, extension, and related topics for functional equations
46L57 Derivations, dissipations and positive semigroups in \(C^*\)-algebras
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[1] Bonsall, F. F.; Duncan, J., Complete Normed Algebras (1973), Springer-Verlag: Springer-Verlag New-York · Zbl 0271.46039
[2] Jewell, N. P.; Sinclair, A. M., Epimorphisms and derivations on \(L^1(0, 1)\) are continuous, Bull. London Math. Soc., 8, 135-139 (1976) · Zbl 0324.46048
[3] A.M. Sinclair, Automatic continuity of linear operators, in: London Math. Soc., in: Lecture Note Series, vol. 21, Cambridge University Press.; A.M. Sinclair, Automatic continuity of linear operators, in: London Math. Soc., in: Lecture Note Series, vol. 21, Cambridge University Press. · Zbl 0313.47029
[4] Ulam, S. M., A Collection of the Mathematical Problems (1960), Interscience Publ.: Interscience Publ. New York · Zbl 0086.24101
[5] Hyers, D. H., On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA, 27, 222-224 (1941) · Zbl 0061.26403
[6] Aoki, T., On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2, 64-66 (1950) · Zbl 0040.35501
[7] Rassias, Th. M., On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72, 297-300 (1978) · Zbl 0398.47040
[8] Gaˇvruta, P., On a generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184, 431-436 (1994) · Zbl 0818.46043
[9] Abbaspour Tabadkan, Gh.; Ramezanpour, M., A fixed point approach to the stability of \(\phi \)-morphisms on the Hilbert \(C^\ast \)-modules, Ann. Funct. Anal., 1, 44-50 (2010) · Zbl 1221.39034
[10] Gajda, Z., On stability of additive mappings, Int. J. Math. Math. Sci., 14, 431-434 (1991) · Zbl 0739.39013
[11] Gaˇvruta, P.; Jung, S.-M.; Li, Y., Hyers-Ulam-Rassias stability of mean value points, Ann. Funct. Anal., 1, 68-74 (2010) · Zbl 1219.39013
[12] Jung, S.-M.; Moslehian, M. S.; Sahoo, P. K., Stability of a generalized Jensen equation on restricted domains, J. Math. Inequal., 4, 191-205 (2010) · Zbl 1219.39016
[13] Rassias, J. M., On approximation of approximately linear mappings by linear mappings, J. Funct. Anal., 46, 126-130 (1982) · Zbl 0482.47033
[14] Rassias, Th. M.; Šemrl, P., On the behavior of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc., 114, 989-993 (1992) · Zbl 0761.47004
[15] Jung, S.-M., Hyers-Ulam-Rassias stability of functional equations in nonlinear analysis, (Springer Optimization and Its Applications, vol. 48 (2011), Springer: Springer New York) · Zbl 1221.39038
[16] Šemrl, P., The functional equation of multiplicative derivation is superstable on standard operator algebras, Integral Equations Operator Theory, 18, 118-122 (1994) · Zbl 0810.47029
[17] Badora, R., On approximate ring homomorphisms, J. Math. Anal. Appl., 276, 589-597 (2002) · Zbl 1014.39020
[18] Bourgin, D. G., Approximately isometric and multiplicative transformations on continuous function rings, Duke Math. J., 16, 385-397 (1949) · Zbl 0033.37702
[19] Badora, R., On approximate derivations, Math. Inequal. Appl., 9, 167-173 (2006) · Zbl 1093.39024
[20] Amyari, M.; Moslehian, M. S., Hyers-Ulam-Rassias stability of derivations on the Hilbert \(C^\ast \)-modules, (Topological Algebras and Applications. Topological Algebras and Applications, Contemp. Math., vol. 427 (2007), Amer. Math. Soc: Amer. Math. Soc Providence, RI), 31-39 · Zbl 1121.39028
[21] Gordji, M. E.; Moslehian, M. S., A trick for investigation of approximate derivations, Math. Commun., 15, 99-105 (2010) · Zbl 1196.39017
[22] Miura, T.; Oka, H.; Hirasawa, G.; Takahasi, S.-E., Sperstability of multipliers and ring derivations on Banach algebras, Banach J. Math., 1, 125-130 (2007) · Zbl 1129.46040
[23] N. Jacobson, Structure of rings, Providence, RI, 1956.; N. Jacobson, Structure of rings, Providence, RI, 1956. · Zbl 0073.02002
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