×

On approximate ternary \(m\)-derivations and \(\sigma\)-homomorphisms. (English) Zbl 1328.39042

Summary: In this paper, we introduce ternary modules over ternary algebras and, using fixed point methods, we prove the stability and superstability of ternary additive, quadratic, cubic and quartic derivations and {\(\sigma\)}-homomorphisms in such structures for the functional equation \[ \begin{aligned} &f(ax+y)+f(ax-y)\\ &\quad= a^{m-2}[f(x+y)+f(x-y)]\\ &\qquad+2(a^{2}-1)\left[a^{m-2}f(x)+\frac{(m-2)(1-(m-2)^{2})}{6}f(y)\right]\end{aligned} \]
for each \(m = 1\), \(2\), \(3\), \(4\).

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B72 Systems of functional equations and inequalities
39B52 Functional equations for functions with more general domains and/or ranges
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] J. Aczél and J. Dhombres, Functional Equations in Several Variables. Cambridge University Press, Cambridge, 1989. · Zbl 0685.39006
[2] Aoki T.: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Japan 2, 64-66 (1950) · Zbl 0040.35501 · doi:10.2969/jmsj/00210064
[3] Badora R.: On approximate derivations. Math. Inequal. Appl. 9, 167-173 (2006) · Zbl 1093.39024
[4] M. Bavand Savadkouhi, M. Eshaghi Gordji, J. M. Rassias and N. Ghobadipour, Approximate ternary Jordan derivations on Banach ternary algebras. J. Math. Phys. 50 (2009), 042303, doi:10.1063/1.3093269, 9 pages. · Zbl 1214.46034
[5] Bazunova N., Borowiec A., Kerner R.: Universal differential calculus on ternary algebras. Lett. Math. Phys. 67, 195-206 (2004) · Zbl 1062.46056 · doi:10.1023/B:MATH.0000035030.12929.cc
[6] Bourgin D. G.: Approximately isometric and multiplicative transformations on continuous function rings. Duke Math. J. 16, 385-397 (1949) · Zbl 0033.37702 · doi:10.1215/S0012-7094-49-01639-7
[7] Bourgin D. G.: Classes of transformations and bordering transformations. Bull. Amer. Math. Soc. 57, 223-237 (1951) · Zbl 0043.32902 · doi:10.1090/S0002-9904-1951-09511-7
[8] L. Cǎdariu and V. Radu, Fixed points and the stability of Jensens functional equation. J. Ineq. Pure Appl. Math. 4 (2003), Article 4, 7 pages, http://jipam.vu.edu.au. · Zbl 1043.39010
[9] L. Cǎdariu and V. Radu, On the stability of the Cauchy functional equation: A fixed point approach. In: Iteration Theory (ECIT ’02), Grazer Math. Ber. 346, Karl-Franzens-Univ. Graz, Graz, 2004, 43-52. · Zbl 1060.39028
[10] A. Ebadian, N. Ghobadipour and M. Eshaghi Gordji, A fixed point method for perturbation of bimultipliers and Jordan bimultipliers in C*-ternary algebras. J. Math. Phys. 51 (2010), doi:10.1063/1.3496391, 10 pages. · Zbl 1314.46063
[11] M. Eshaghi Gordji, Z. Alizadeh, H. Khodaei and C. Park, On approximate homomorphisms: A fixed point approach. Math. Sci. 6 (2012), doi:10.1186/2251-7456-6-59, 8 pages. · Zbl 1271.39023
[12] Eshaghi Gordji M., Ghaemi M. B., Kaboli Gharetapeh S., Shams S., Ebadian A.: On the stability of J*-derivations. J. Geom. Phys. 60, 454-459 (2010) · Zbl 1188.39029 · doi:10.1016/j.geomphys.2009.11.004
[13] M. Eshaghi Gordji, S. Kaboli Gharetapeh, E. Rashidi, T. Karimi, and M. Aghaei, Ternary Jordan* derivations on C*-ternary algebras. J. Comput. Anal. Appl. 12 (2010), 463-470. · Zbl 1198.39038
[14] M. Eshaghi Gordji and H. Khodaei, On the generalized Hyers-Ulam-Rassias stability of quadratic functional equations. Abstr. Appl. Anal. 2009 (2009), Article ID 923476, 11 pages. · Zbl 1167.39014
[15] Eshaghi Gordji M., Khodaei H.: Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces. Nonlinear Anal. 71, 5629-5643 (2009) · Zbl 1179.39034 · doi:10.1016/j.na.2009.04.052
[16] M. Eshaghi Gordji and H. Khodaei, The fixed point method for fuzzy approximation of a functional equation associated with inner product spaces. Discrete Dyn. Nat. Soc. 2010 (2010), Article ID 140767, doi:10.1155/2010/140767, 15 pages. · Zbl 1221.39036
[17] Forti G.-L.: Elementary remarks on Ulam-Hyers stability of linear functional equations. J. Math. Anal. Appl. 328, 109-118 (2007) · Zbl 1111.39026 · doi:10.1016/j.jmaa.2006.04.079
[18] Gajda Z.: On stability of additive mappings. Int. J. Math. Math. Sci. 14, 431-434 (1991) · Zbl 0739.39013 · doi:10.1155/S016117129100056X
[19] Gǎvruţa P.: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431-436 (1994) · Zbl 0818.46043 · doi:10.1006/jmaa.1994.1211
[20] Hyers D. H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27, 222-224 (1941) · Zbl 0061.26403 · doi:10.1073/pnas.27.4.222
[21] D. H. Hyers, G. Isac and Th. M. Rassias, Stability of Functional Equations in Several Variables. Birkhauser Boston, Boston, MA, 1998. · Zbl 0907.39025
[22] Isac G., Rassias Th. M.: On the Hyers-Ulam stability of ψ-additive mappings. J. Approx. Theory 72, 131-137 (1993) · Zbl 0770.41018 · doi:10.1006/jath.1993.1010
[23] Jun K.-W., Kim H.-M.: The generalized Hyers-Ulam-Rassias stability of a cubic functional equation. J. Math. Anal. Appl. 274, 267-278 (2002) · Zbl 1021.39014 · doi:10.1016/S0022-247X(02)00415-8
[24] S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press Inc., Palm Harbor, FL, 2001. · Zbl 0980.39024
[25] Khodaei H., Rassias Th. M.: Approximately generalized additive functions in several variables. Int. J. Nonlinear Anal. Appl. 1, 22-41 (2010) · Zbl 1281.39041
[26] Lee S .H., Im S. M., Hawng I. S.: Quartic functional equation. J. Math. Anal. Appl. 307, 387-394 (2005) · Zbl 1072.39024 · doi:10.1016/j.jmaa.2004.12.062
[27] Moslehian M. S.: Almost derivations on C*-ternary rings. Bull Belg. Math. Soc. Simon Stevin 14, 135-142 (2007) · Zbl 1132.39026
[28] Park C.: Homomorphisms between Lie JC*-algebras and Cauchy-Rassias stability of Lie JC*-algebra derivations. J. Lie Theory 15, 393-414 (2005) · Zbl 1091.39006
[29] C. Park and J. Cui, Generalized stability of C*-ternary quadratic mappings. Abstr. Appl. Anal. 2007 (2007), Article ID 23282, 6 pages. · Zbl 1158.39020
[30] C. Park and M. Eshaghi Gordji, Comment on “Approximate ternary Jordan derivations on Banach ternary algebras” [Bavand Savadkouhi et al., J. Math. Phys. 50, 042303 (2009)]. J. Math. Phys. 51 (2010), 044102, doi:10.1063/1.3299295, 7 pages. · Zbl 1310.46047
[31] Park C., Rassias Th. M.: Isomorphisms in unital C*-algebras. Int. J. Nonlinear Anal. Appl. 1, 1-10 (2010) · Zbl 1281.39027
[32] Rassias J. M.: On approximation of approximately linear mappings by linear mappings. J. Funct. Anal. 46, 126-130 (1982) · Zbl 0482.47033 · doi:10.1016/0022-1236(82)90048-9
[33] Rassias J.M.: On approximation of approximately linear mappings by linear mappings. Bull. Sci. Math. 108, 445-446 (1984) · Zbl 0599.47106
[34] Rassias Th.M.: On the stability of the linear mapping in Banach spaces. Proc. Amer. Math. Soc. 72, 297-300 (1978) · Zbl 0398.47040 · doi:10.1090/S0002-9939-1978-0507327-1
[35] 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 · doi:10.1090/S0002-9939-1992-1059634-1
[36] I. A. Rus, Principles and Applications of Fixed Point Theory. Dacia, Cluj-Napoca, 1979.
[37] 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 · doi:10.1007/BF01225216
[38] Shagholi S., Eshagi Gordji M., Bavand Savadkouhi M.: Stability of ternary quadratic derivation on ternary Banach algebras. J. Comput. Anal. Appl. 13, 1097-1105 (2011) · Zbl 1225.39030
[39] S. M. Ulam, Problems in Modern Mathematics. Science Editions, Wiley, New York, 1964. · Zbl 0137.24201
[40] E. Zeidler, Nonlinear Functional Analysis and Its Applications. I. Fixed-Point Theorems. 2nd corrected printing, Springer-Verlag, New York, 1993. · Zbl 0794.47033
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.