# zbMATH — the first resource for mathematics

Permuting triderivations and permuting trihomomorphisms in Banach algebras. (English) Zbl 1452.39006
Summary: Using the direct method, we prove the Hyers-Ulam stability of permuting triderivations and permuting trihomomorphisms in Banach algebras and unital $$C^\ast$$-algebras associated with the triadditive $$s$$-functional inequality \begin{multline*} \| f (x+y, z-w, a+b)+f(x-y, z+w, a-b)-2f (x, z, a)+2f(x, w, b)-2f(y, z, b)+2f(y, w, a) \| \\ \leq \| s (2f((x+y) / 2, z-w, a+b)+2f((x-y) / 2, z+w, a-b) \\ -2f(x, z, a)+ 2f(x, w, b)-2f (y, z, b)+2f(y, w, a)) \|, \end{multline*} where $$s$$ is a fixed nonzero complex number with $$|s|< 1$$.
##### MSC:
 39B52 Functional equations for functions with more general domains and/or ranges 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 47B47 Commutators, derivations, elementary operators, etc. 17A40 Ternary compositions
Full Text:
##### References:
  T. Aoki, “On the stability of the linear transformation in Banach spaces”, J. Math. Soc. Japan 2:1-2 (1950), 64-66. Mathematical Reviews (MathSciNet): MR40580 Zentralblatt MATH: 0040.35501 Digital Object Identifier: doi:10.2969/jmsj/00210064 Project Euclid: euclid.jmsj/1261735234 · Zbl 0040.35501  J.-H. Bae and W.-G. Park, “Approximate bi-homomorphisms and bi-derivations in $$C^*$$-ternary algebras”, Bull. Korean Math. Soc. 47:1 (2010), 195-209. Mathematical Reviews (MathSciNet): MR2604245 Zentralblatt MATH: 1188.39026 Digital Object Identifier: doi:10.4134/BKMS.2010.47.1.195 · Zbl 1188.39026  N. Eghbali, J. M. Rassias, and M. Taheri, “On the stability of a $$k$$-cubic functional equation in intuitionistic fuzzy $$n$$-normed spaces”, Results Math. 70:1-2 (2016), 233-248. Mathematical Reviews (MathSciNet): MR3535004 Zentralblatt MATH: 1360.39022 Digital Object Identifier: doi:10.1007/s00025-015-0476-9 · Zbl 1360.39022  I.-i. El-Fassi, J. Brzdęk, A. Chahbi, and S. Kabbaj, “On hyperstability of the biadditive functional equation”, Acta Math. Sci. Ser. $$B ($$ Engl. Ed.$$) 37$$:6 (2017), 1727-1739. Mathematical Reviews (MathSciNet): MR3706720 Zentralblatt MATH: 1399.39066 Digital Object Identifier: doi:10.1016/S0252-9602(17)30103-0 · Zbl 1399.39066  M. Eshaghi Gordji and A. Fazeli, “Stability and superstability of homomorphisms on $$C^*$$-ternary algebras”, An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat. 20:1 (2012), 173-187. Mathematical Reviews (MathSciNet): MR2928416 Zentralblatt MATH: 1274.39054 · Zbl 1274.39054  G. Z. Eskandani and J. M. Rassias, “Approximation of general $$\alpha$$-cubic functional equations in 2-Banach spaces”, Ukraïn. Mat. Zh. 68:10 (2016), 1430-1436. Mathematical Reviews (MathSciNet): MR3570430 · Zbl 07030309  M. Fošner and J. Vukman, “On some functional equations arising from $$(m,n)$$-Jordan derivations and commutativity of prime rings”, Rocky Mountain J. Math. 42:4 (2012), 1153-1168. Mathematical Reviews (MathSciNet): MR2981038 Zentralblatt MATH: 1262.16044 Digital Object Identifier: doi:10.1216/RMJ-2012-42-4-1153 Project Euclid: euclid.rmjm/1348752079 · Zbl 1262.16044  P. Găvruţa, “A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings”, J. Math. Anal. Appl. 184:3 (1994), 431-436. Mathematical Reviews (MathSciNet): MR1281518 Zentralblatt MATH: 0818.46043 Digital Object Identifier: doi:10.1006/jmaa.1994.1211 · Zbl 0818.46043  D. H. Hyers, “On the stability of the linear functional equation”, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224. Mathematical Reviews (MathSciNet): MR4076 Zentralblatt MATH: 0061.26403 Digital Object Identifier: doi:10.1073/pnas.27.4.222 · Zbl 0061.26403  R. V. Kadison and J. R. Ringrose, Fundamentals of the theory of operator algebras: elementary theory, Pure and Applied Mathematics 100, Academic Press, New York, 1983. Mathematical Reviews (MathSciNet): MR719020 Zentralblatt MATH: 0518.46046 · Zbl 0518.46046  H. Khodaei, “On the stability of additive, quadratic, cubic and quartic set-valued functional equations”, Results Math. 68:1-2 (2015), 1-10. Mathematical Reviews (MathSciNet): MR3391489 Zentralblatt MATH: 1330.39029 Digital Object Identifier: doi:10.1007/s00025-014-0416-0 · Zbl 1330.39029  D. Özden, M. A. Öztürk, and Y. B. Jun, “Permuting tri-derivations in prime and semi-prime gamma rings”, Kyungpook Math. J. 46:2 (2006), 153-167. Mathematical Reviews (MathSciNet): MR2235554 Zentralblatt MATH: 1162.16317 · Zbl 1162.16317  M. A. Öztürk, “Permuting tri-derivations in prime and semi-prime rings”, East Asian Math. J. 15:2 (1999), 177-190. Zentralblatt MATH: 1162.16317 · Zbl 1162.16317  C. Park, “Additive $$\rho$$-functional inequalities and equations”, J. Math. Inequal. 9:1 (2015), 17-26. Mathematical Reviews (MathSciNet): MR3333901 Zentralblatt MATH: 1314.39026 Digital Object Identifier: doi:10.7153/jmi-09-02 · Zbl 1314.39026  C. Park, “Additive $$\rho$$-functional inequalities in non-Archimedean normed spaces”, J. Math. Inequal. 9:2 (2015), 397-407. Mathematical Reviews (MathSciNet): MR3333870 Zentralblatt MATH: 1323.39023 Digital Object Identifier: doi:10.7153/jmi-09-33 · Zbl 1323.39023  C. Park, “Permuting triderivations and permuting trihomomorphisms in complex Banach algebras”, preprint, 2020. arXiv: 2009.10346  T. M. Rassias, “On the stability of the linear mapping in Banach spaces”, Proc. Amer. Math. Soc. 72:2 (1978), 297-300. Mathematical Reviews (MathSciNet): MR507327 Zentralblatt MATH: 0398.47040 Digital Object Identifier: doi:10.1090/S0002-9939-1978-0507327-1 · Zbl 0398.47040  I. B. Risteski, “Some higher order complex vector functional equations”, Czechoslovak Math. J. 54(129):4 (2004), 1015-1034. Mathematical Reviews (MathSciNet): MR2100011 Zentralblatt MATH: 1080.39506 Digital Object Identifier: doi:10.1007/s10587-004-6448-y · Zbl 1080.39506  S. M. Ulam, A collection of mathematical problems, Interscience Tracts in Pure and Applied Mathematics 8, Interscience Publishers, New York, 1960. Mathematical Reviews (MathSciNet): MR0120127 Zentralblatt MATH: 0086.24101 · Zbl 0086.24101  H. Yazarli, “Permuting triderivations of prime and semiprime rings”, Miskolc Math. Notes 18:1 (2017), 489-497. Mathematical Reviews (MathSciNet): MR3666527 Zentralblatt MATH: 1399.16052 Digital Object Identifier: doi:10.18514/MMN.2017.1647 · Zbl 1399.16052  H. · Zbl 1109.16303
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.