Approximate ternary Jordan derivations on Banach ternary algebras. (English) Zbl 1214.46034
Summary: Let \(A\) be a Banach ternary algebra over a scalar field \(\mathbb R\) or \(\mathbb C\) and \(X\) be a ternary Banach \(A\)-module. A linear mapping \(D: (A,[\,]_ A)\to(X,[\,]_ X)\) is called a ternary Jordan derivation if \(D([xxx]_ A)=[D(x)xx]_ X+[xD(x)x]_ X+[xxD(x)]_ X\) for all \(x\in A\). In this paper, we investigate ternary Jordan derivations on Banach ternary algebras, associated with the following functional equation \(f((x+y+z)/4)+f((3x-y-4z)/4)+f((4x+3z)/4)=2f(x)\). Moreover, we prove the generalized Ulam-Hyers stability of ternary Jordan derivations on Banach ternary algebras.
In a comment C.-G. Park and M. Eshaghi Gordji [J. Math. Phys. 51, No. 4, 044102 (2010)] state: The mapping \(f\) in Lemma 2.2 of B. Savadkouhi et al. is identically zero and all of the results are trivial. In this note, we correct the statements of the results and the proofs.
In a comment C.-G. Park and M. Eshaghi Gordji [J. Math. Phys. 51, No. 4, 044102 (2010)] state: The mapping \(f\) in Lemma 2.2 of B. Savadkouhi et al. is identically zero and all of the results are trivial. In this note, we correct the statements of the results and the proofs.
MSC:
| 46K05 | General theory of topological algebras with involution |
| 39B82 | Stability, separation, extension, and related topics for functional equations |
| 47B47 | Commutators, derivations, elementary operators, etc. |
References:
| [1] | DOI: 10.2307/2369145 · JFM 13.0107.01 · doi:10.2307/2369145 |
| [2] | Kapranov M., Discrimininants, Resultants and Multidimensional Determinants (1994) · Zbl 0827.14036 |
| [3] | DOI: 10.1063/1.531821 · Zbl 0872.58006 · doi:10.1063/1.531821 |
| [4] | DOI: 10.1007/s11005-005-0042-6 · Zbl 1112.39021 · doi:10.1007/s11005-005-0042-6 |
| [5] | Amyari M., Taiwan. J. Math. 11 pp 1417– (2007) |
| [6] | DOI: 10.1016/j.na.2005.04.004 · Zbl 1085.39026 · doi:10.1016/j.na.2005.04.004 |
| [7] | DOI: 10.1007/BF01055705 · Zbl 0892.58090 · doi:10.1007/BF01055705 |
| [8] | DOI: 10.1023/B:MATH.0000035030.12929.cc · Zbl 1062.46056 · doi:10.1023/B:MATH.0000035030.12929.cc |
| [9] | DOI: 10.1063/1.1704187 · Zbl 0139.46003 · doi:10.1063/1.1704187 |
| [10] | R. Kerner, ”Ternary algebraic structures and their applications in physics,” (Pierre et Marie Curie University, Paris, 2000). |
| [11] | DOI: 10.1088/0264-9381/14/1A/017 · Zbl 0897.17002 · doi:10.1088/0264-9381/14/1A/017 |
| [12] | Moslehian M. S., Bull. Belg. Math. Soc. Simon Stevin 14 pp 135– (2007) |
| [13] | DOI: 10.1007/s10440-007-9179-x · Zbl 1135.39014 · doi:10.1007/s10440-007-9179-x |
| [14] | Sewell G. L., Quantum Mechanics and its Emergent Macrophysics (2002) · Zbl 1007.82001 |
| [15] | DOI: 10.1007/BF02103278 · Zbl 0808.70015 · doi:10.1007/BF02103278 |
| [16] | DOI: 10.1063/1.531526 · Zbl 0864.17002 · doi:10.1063/1.531526 |
| [17] | DOI: 10.1016/0001-8708(83)90083-X · Zbl 0517.46049 · doi:10.1016/0001-8708(83)90083-X |
| [18] | Ulam S. M., Problems in Modern Mathematics (1940) · Zbl 0137.24201 |
| [19] | DOI: 10.1073/pnas.27.4.222 · Zbl 0061.26403 · doi:10.1073/pnas.27.4.222 |
| [20] | DOI: 10.2969/jmsj/00210064 · Zbl 0040.35501 · doi:10.2969/jmsj/00210064 |
| [21] | DOI: 10.1215/S0012-7094-49-01639-7 · Zbl 0033.37702 · doi:10.1215/S0012-7094-49-01639-7 |
| [22] | DOI: 10.2307/2042795 · Zbl 0398.47040 · doi:10.2307/2042795 |
| [23] | Rassias J. M., Bull. Sci. Math. 108 pp 445– (1984) |
| [24] | DOI: 10.1016/0022-1236(82)90048-9 · Zbl 0482.47033 · doi:10.1016/0022-1236(82)90048-9 |
| [25] | Rassias J. M., Glas. Mat. Ser. III 34 pp 243– (1999) |
| [26] | Rassias J. M., Discuss. Math. 14 pp 101– (1994) |
| [27] | J. M. Rassias, Geometry, Analysis and Mechanics (World Scientific, River Edge, NJ, 1994), pp. 365–375. |
| [28] | DOI: 10.1142/9789814360166_0019 · doi:10.1142/9789814360166_0019 |
| [29] | Rassias J. M., Discuss. Math. 12 pp 95– (1992) |
| [30] | DOI: 10.1016/0021-9045(89)90041-5 · Zbl 0672.41027 · doi:10.1016/0021-9045(89)90041-5 |
| [31] | Rassias J. M., Discuss. Math. 7 pp 193– (1985) |
| [32] | DOI: 10.1063/1.2942415 · Zbl 1152.81589 · doi:10.1063/1.2942415 |
| [33] | DOI: 10.1007/BF02192660 · Zbl 0549.39006 · doi:10.1007/BF02192660 |
| [34] | Czerwik S., Stability of Functional Equations of Ulam-Hyers-Rassias Type (2003) · Zbl 1382.39036 |
| [35] | DOI: 10.1155/S016117129100056X · Zbl 0739.39013 · doi:10.1155/S016117129100056X |
| [36] | DOI: 10.1007/978-1-4612-1790-9 · doi:10.1007/978-1-4612-1790-9 |
| [37] | Rassias Th. M., Math. Appl. 62 pp 23– (2000) |
| [38] | DOI: 10.1006/jmaa.2000.7046 · Zbl 0964.39026 · doi:10.1006/jmaa.2000.7046 |
| [39] | DOI: 10.1006/jmaa.2000.6788 · Zbl 0958.46022 · doi:10.1006/jmaa.2000.6788 |
| [40] | Ravi K., Int. J. Math. Stat. 3 pp 36– (2008) |
| [41] | Badora R., Math. Inequal. Appl. 9 pp 167– (2006) |
| [42] | DOI: 10.1016/j.jmaa.2005.06.060 · Zbl 1104.39025 · doi:10.1016/j.jmaa.2005.06.060 |
| [43] | DOI: 10.1006/jath.1993.1010 · Zbl 0770.41018 · doi:10.1006/jath.1993.1010 |
| [44] | DOI: 10.1016/j.jmaa.2006.04.010 · Zbl 1113.39033 · doi:10.1016/j.jmaa.2006.04.010 |
| [45] | DOI: 10.3836/tjm/1270130262 · Zbl 0801.46060 · doi:10.3836/tjm/1270130262 |
| [46] | DOI: 10.1007/s00574-005-0029-z · Zbl 1091.39007 · doi:10.1007/s00574-005-0029-z |
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.
