×

Survey on recent Ulam stability results concerning derivations. (English) Zbl 1347.39025

Summary: This is a survey presenting the most significant results concerning approximate (generalized) derivations, motivated by the notions of Ulam and Hyers-Ulam stability. Moreover, the hyperstability and superstability issues connected with derivations are discussed. In the section before the last one we highlight some recent outcomes on stability of conditions defining (generalized) Lie derivations.

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
46L57 Derivations, dissipations and positive semigroups in \(C^*\)-algebras
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ulam, S. M., A Collection of Mathematical Problems (1960), New York, NY, USA: Interscience Publishers, New York, NY, USA · Zbl 0086.24101
[2] Ulam, S. M., Problems in Modern Mathematics (1964), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0137.24201
[3] Hyers, D. H., On the stability of the linear functional equation, Proceedings of the National Academy of Sciences of the United States of America, 27, 222-224 (1941) · Zbl 0061.26403
[4] Aoki, T., On the stability of the linear transformation in Banach spaces, Journal of the Mathematical Society of Japan, 2, 64-66 (1950) · Zbl 0040.35501
[5] Rassias, T. M., On the stability of the linear mapping in Banach spaces, Proceedings of the American Mathematical Society, 72, 2, 297-300 (1978) · Zbl 0398.47040
[6] Brzdęk, J.; Fechner, W.; Moslehian, M. S.; Sikorska, J., Recent developments of the conditional stability of the homomorphism equation, Banach Journal of Mathematical Analysis, 9, 3, 278-326 (2015) · Zbl 1312.39031
[7] Hyers, D. H.; Isac, G.; Rassias, T. M., Stability of Functional Equations in Several Variables. Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and their Applications, 34 (1998), Boston, Mass, USA: Birkhäuser Boston, Boston, Mass, USA · Zbl 0907.39025
[8] Jung, S.-M., Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis. Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer Optimization and Its Applications, 48 (2011), New York, NY, USA: Springer, New York, NY, USA · Zbl 1221.39038
[9] Rassias, T. M., Functional Equations, Inequalities and Applications (2003), Dordrecht, The Netherlands: Kluwer Academic, Dordrecht, The Netherlands · Zbl 1047.39001
[10] Brešar, M., On the distance of the composition of two derivations to the generalized derivations, Glasgow Mathematical Journal, 33, 1, 89-93 (1991) · Zbl 0731.47037
[11] Mathieu, M., Elementary Operators & Applications. In Memory of Domingo A. Herrero. Proceedings of the International Workshop held in Blaubeuren, June 912, 1991 (1992), River Edge, NJ, USA: World Scientific, River Edge, NJ, USA
[12] Hvala, B., Generalized derivations in rings, Communications in Algebra, 26, 4, 1147-1166 (1998) · Zbl 0899.16018
[13] Vukman, J., A note on generalized derivations of semiprime rings, Taiwanese Journal of Mathematics, 11, 2, 367-370 (2007) · Zbl 1124.16030
[14] Singer, I. M.; Wermer, J., Derivations on commutative normed algebras, Mathematische Annalen, 129, 260-264 (1955) · Zbl 0067.35101
[15] Thomas, M. P., The image of a derivation is contained in the radical, Annals of Mathematics, 128, 3, 435-460 (1988) · Zbl 0681.47016
[16] Johnson, B. E., Continuity of derivations on commutative algebras, American Journal of Mathematics, 91, 1-10 (1969) · Zbl 0181.41103
[17] Hatori, O.; Wada, J., Ring derivations on semi-simple commutative Banach algebras, Tokyo Journal of Mathematics, 15, 1, 223-229 (1992) · Zbl 0801.46060
[18] Miura, T.; Hirasawa, G.; Takahasi, S.-E., A perturbation of ring derivations on Banach algebras, Journal of Mathematical Analysis and Applications, 319, 2, 522-530 (2006) · Zbl 1104.39025
[19] Šemrl, P., The functional equation of multiplicative derivation is superstable on standard operator algebras, Integral Equations and Operator Theory, 18, 1, 118-122 (1994) · Zbl 0810.47029
[20] Jun, K.-W.; Park, D.-W., Almost derivations on the Banach algebra \(C^n \left``[0, 1\right``]\), Bulletin of the Korean Mathematical Society, 33, 3, 359-366 (1996)
[21] Johnson, B. E., Approximately multiplicative functionals, Journal of the London Mathematical Society. Second Series, 34, 3, 489-510 (1986) · Zbl 0625.46059
[22] Johnson, B. E., Approximately multiplicative maps between Banach algebras, Journal of the London Mathematical Society. Second Series, 37, 2, 294-316 (1988) · Zbl 0652.46031
[23] Jarosz, K., Perturbations of Banach Algebras (1985), Berlin, Germany: Springer, Berlin, Germany · Zbl 0557.46029
[24] Jarosz, K., Ultraproducts and small bound perturbations, Pacific Journal of Mathematics, 148, 1, 81-88 (1991) · Zbl 0755.46005
[25] Amyari, M.; Moslehian, M. S., Hyers-Ulam-Rassias stability of derivations on Hilbert \(C^∗\)-modules, Topological Algebras and Applications. Topological Algebras and Applications, Contemporary Mathematics, 427, 31-39 (2007), Providence, RI, USA: American Mathematical Society, Providence, RI, USA · Zbl 1121.39028
[26] Badora, R., On approximate derivations, Mathematical Inequalities & Applications, 9, 1, 167-173 (2006) · Zbl 1093.39024
[27] Gordji, M. E.; Moslehian, M. S., A trick for investigation of approximate derivations, Mathematical Communications, 15, 1, 99-105 (2010) · Zbl 1196.39017
[28] Park, C.-G., Linear derivations on Banach algebras, Nonlinear Functional Analysis and Applications, 9, 3, 359-368 (2004) · Zbl 1068.47039
[29] Park, C.-G., Lie ∗-homomorphisms between Lie \(C^∗\)-algebras and Lie ∗-derivations on Lie \(C^∗\)-algebras, Journal of Mathematical Analysis and Applications, 293, 2, 419-434 (2004) · Zbl 1051.46052
[30] Amyari, M.; Baak, C.; Moslehian, M. S., Nearly ternary derivations, Taiwanese Journal of Mathematics, 11, 5, 1417-1424 (2007) · Zbl 1141.39024
[31] Polat, F., Approximate Riesz algebra-valued derivations, Abstract and Applied Analysis, 2012 (2012) · Zbl 1264.46035
[32] Isac, G.; Rassias, T. M., On the Hyers-Ulam stability of \(ψ\)-additive mappings, Journal of Approximation Theory, 72, 2, 131-137 (1993) · Zbl 0770.41018
[33] Gajda, Z., On stability of additive mappings, International Journal of Mathematics and Mathematical Sciences, 14, 3, 431-434 (1991) · Zbl 0739.39013
[34] Moslehian, M. S., Hyers-Ulam-Rassias stability of generalized derivations, International Journal of Mathematics and Mathematical Sciences, 2006 (2006) · Zbl 1120.39029
[35] Badora, R., On approximate ring homomorphisms, Journal of Mathematical Analysis and Applications, 276, 2, 589-597 (2002) · Zbl 1014.39020
[36] Dales, H. G., Banach Algebras and Automatic Continuity (2000), New York, NY, USA: The Clarendon Press, Oxford University Press, New York, NY, USA · Zbl 0981.46043
[37] Gordji, M. E.; Filali, M., Arens regularity of module actions, Studia Mathematica, 181, 3, 237-254 (2007) · Zbl 1165.46024
[38] Fošner, A.; Moslehian, M. S., On approximate generalized derivations, The Natália Bebiano Anniversary Volume. The Natália Bebiano Anniversary Volume, Textos de Matematica Série B, 44, 33-44 (2013), Coimbra, Portugal: University of Coimbra, Coimbra, Portugal · Zbl 1307.46039
[39] Baker, J. A., The stability of the cosine equation, Proceedings of the American Mathematical Society, 80, 3, 411-416 (1980) · Zbl 0448.39003
[40] Jung, S.-M.; Rassias, M. T.; Mortici, C., On a functional equation of trigonometric type, Applied Mathematics and Computation, 252, 294-303 (2015) · Zbl 1338.39041
[41] Moszner, Z., Sur les définitions différentes de la stabilité des équations fonctionnelles, Aequationes Mathematicae, 68, 3, 260-274 (2004) · Zbl 1060.39031
[42] Bourgin, D. G., Approximately isometric and multiplicative transformations on continuous function rings, Duke Mathematical Journal, 16, 385-397 (1949) · Zbl 0033.37702
[43] Maksa, Gy.; Páles, Zs., Hyperstability of a class of linear functional equations, Acta Mathematica Academiae Paedagogiace Nyíregyháziensis, New Series, 17, 2, 107-112 (2001) · Zbl 1004.39022
[44] Gselmann, E., Hyperstability of a functional equation, Acta Mathematica Hungarica, 124, 1-2, 179-188 (2009) · Zbl 1212.39044
[45] Maksa, Gy.; Nikodem, K.; Páles, Zs., Results on \(t\)-Wright convexity, Comptes Rendus Mathématiques des l’Académie des Sciences. La Société Royale du Canada, 13, 6, 274-278 (1991) · Zbl 0749.26007
[46] Brillouët-Belluot, N.; Brzdęk, J.; Ciepliński, K., On some recent developments in Ulam’s type stability, Abstract and Applied Analysis, 2012 (2012) · Zbl 1259.39019
[47] Brzdęk, J.; Ciepliński, K., Hyperstability and superstability, Abstract and Applied Analysis, 2013 (2013) · Zbl 1293.39013
[48] Cao, H.-X.; Lv, J.-R.; Rassias, J. M., Superstability for generalized module left derivations and generalized module derivations on a Banach module (I), Journal of Inequalities and Applications, 2009 (2009) · Zbl 1185.46033
[49] Cao, H.-X.; Lv, J.-R.; Rassias, J. M., Superstability for generalized module left derivations and generalized module derivations on a Banach module. II, Journal of Inequalities in Pure and Applied Mathematics, 10, 3, article 85 (2009) · Zbl 1211.39015
[50] Fošner, A., On the generalized Hyers-Ulam stability of module left ( m, n)-derivations, Aequationes Mathematicae, 84, 1-2, 91-98 (2012) · Zbl 1267.39022
[51] Fošner, A., Hyers-Ulam-Rassias stability of generalized module left ( m, n)-derivations, Journal of Inequalities and Applications, 2013, article 2013:208 (2013) · Zbl 1282.39028
[52] Moslehian, M. S., Ternary derivations, stability and physical aspects, Acta Applicandae Mathematicae, 100, 2, 187-199 (2008) · Zbl 1135.39014
[53] Moslehian, M. S., Superstability of higher derivations in multi-Banach algebras, Tamsui Oxford Journal of Mathematical Sciences, 24, 4, 417-427 (2008) · Zbl 1175.39022
[54] Park, C.-G.; Gordji, M. E.; Cho, Y. J., Stability and superstability of generalized quadratic ternary derivations on non-Archimedean ternary Banach algebras: a fixed point approach, Fixed Point Theory and Applications, 2012, article 2012:97 (2012) · Zbl 1277.39044
[55] Shateri, T. L., Superstability of generalized higher derivations, Abstract and Applied Analysis, 2011 (2011) · Zbl 1228.39030
[56] Kuczma, M., An Introduction to the Theory of Functional Equations and Inequalities. Cauchy’s Equation and Jensen’s Inequality (2009), Basel, Switzerland: Birkhäuser, Basel, Switzerland · Zbl 1221.39041
[57] Jabłoński, W., On a class of sets connected with a convex function, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 69, 205-210 (1999) · Zbl 0952.39009
[58] Jabłoński, W., Sum of graphs of continuous functions and boundedness of additive operators, Journal of Mathematical Analysis and Applications, 312, 2, 527-534 (2005) · Zbl 1076.26007
[59] Miura, T.; Oka, H.; Hirasawa, G.; Takahasi, S.-E., Superstability of multipliers and ring derivations on Banach algebras, Banach Journal of Mathematical Analysis, 1, 1, 125-130 (2007) · Zbl 1129.46040
[60] Najati, A.; Park, C., Stability of homomorphisms and generalized derivations on Banach algebras, Journal of Inequalities and Applications, 2009 (2009) · Zbl 1187.39046
[61] Brzdęk, J.; Fošner, A., On approximate generalized Lie derivations, Glasnik Matematicki. Serija III, 50, 1, 77-99 (2015) · Zbl 1321.47088
[62] Ashraf, M.; Al-Shammakh, W., On generalized \((θ,ϕ)\)-derivations in rings, International Journal of Mathematics, Game Theory and Algebra, 12, 4, 295-300 (2002) · Zbl 1076.16512
[63] Mirzavaziri, M.; Moslehian, M. S., Automatic continuity of \(σ\)-derivations on \(C^∗\)-algebras, Proceedings of the American Mathematical Society, 134, 11, 3319-3327 (2006) · Zbl 1116.46061
[64] Amyari, M.; Rahbarnia, F.; Sadeghi, G., Some results on stability of extended derivations, Journal of Mathematical Analysis and Applications, 329, 2, 753-758 (2007) · Zbl 1153.39308
[65] Baak, C.; Moslehian, M. S., On the stability of \(\theta \)-derivations on \(J B^*\)-triples, Bulletin of the Brazilian Mathematical Society, 38, 1, 115-127 (2007) · Zbl 1127.39053
[66] Fošner, A., The Hyers-Ulam-Rassias stability of \((m, n)_{(\sigma, \tau)}\)-derivations on normed algebras, Abstract and Applied Analysis, 2012 (2012) · Zbl 1251.39023
[67] Moslehian, M. S., Approximate \(( \sigma, \tau )\)-contractibility, Nonlinear Functional Analysis and Applications, 11, 5, 805-813 (2006) · Zbl 1120.39028
[68] Schwaiger, J., On the stability of derivations of higher order, Annales Academiae Paedagogicae Cracoviensis Studia Mathematica, 4, 139-146 (2001) · Zbl 1140.39322
[69] Gselmann, E., Approximate derivations of order \(n\), Acta Mathematica Hungarica, 144, 1, 217-226 (2014) · Zbl 1340.39042
[70] Jung, Y.-S., On the generalized Hyers-Ulam stability of module left derivations, Journal of Mathematical Analysis and Applications, 339, 1, 108-114 (2008) · Zbl 1134.39023
[71] 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, Taiwanese Journal of Mathematics, 7, 4, 641-655 (2003) · Zbl 1073.39019
[72] Jun, K.-W.; Kim, H.-M., Approximate derivations mapping into the radicals of Banach algebras, Taiwanese Journal of Mathematics, 11, 1, 277-288 (2007) · Zbl 1165.39024
[73] Park, C.; Boo, D.-H., Isomorphisms and generalized derivations in proper \(CQ^∗\)-algebras, Journal of Nonlinear Science and Its Applications, 4, 1, 19-36 (2011) · Zbl 1207.39040
[74] Park, C.; Boo, D.-H.; An, J. S., Homomorphisms between \(C^*\)-algebras and linear derivations on \(C^*\)-algebras, Journal of Mathematical Analysis and Applications, 337, 2, 1415-1424 (2008) · Zbl 1147.39011
[75] Roh, J.; Chang, I.-S., Approximate derivations with the radical ranges of noncommutative Banach algebras, Abstract and Applied Analysis, 2015 (2015) · Zbl 1343.47045
[76] Shateri, T. L.; Sadeghi, G., Stability of derivations in modular spaces, Mediterranean Journal of Mathematics, 11, 3, 929-938 (2014) · Zbl 1308.39027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.