# APPROXIMATELY PARTIAL TERNARY QUADRATIC DERIVATIONS ON BANACH TERNARY ALGEBRAS

Volume 4, Issue 1, pp 60-69
• 1066 Views

### Authors

A. JAVADIAN - Department of Physics, Semnan University, P. O. Box 35195-363, Semnan, Iran. M. ESHAGHI GORDJI - Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran. M. BAVAND SAVADKOUHI - Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran.

### Abstract

Let $A_1,A_2,...,A_n$ be normed ternary algebras over the complex field $\mathbb{C}$ and let $B$ be a Banach ternary algebra over $\mathbb{C}$. A mapping $\delta_k$ from $A_1 \times ...\times A_n$ into $B$ is called a k-th partial ternary quadratic derivation if there exists a mapping $g_k : A_k \rightarrow B$ such that $\delta_k(x_1,..., [a_kb_kc_k],..., x_n) =[g_k(a_k)g_k(b_k)\delta_k(x_1 ,..., c_k,..., xn)] + [g_k(a_k)\delta_k(x_1,..., b_k,..., x_n)g_k(c_k)] + [\delta_k(x_1,...,a_k,..., x_n)g_k(b_k)g_k(c_k)]$ and $\delta_k(x_1,..., a_k + b_k,..., x_n) + \delta_k(x_1,... a_k - b_k,..., x_n) = 2\delta_k(x_1,..., a_k,..., x_n) + 2\delta_k(x_1,...,b_k,..., x_n)$ for all $a_k, b_k, c_k \in A_k$ and all $x_i \in A_i (i \neq k)$. We prove the Hyers-Ulam- Rassias stability of the partial ternary quadratic derivations in Banach ternary algebras.

### Share and Cite

##### ISRP Style

A. JAVADIAN, M. ESHAGHI GORDJI, M. BAVAND SAVADKOUHI, APPROXIMATELY PARTIAL TERNARY QUADRATIC DERIVATIONS ON BANACH TERNARY ALGEBRAS, Journal of Nonlinear Sciences and Applications, 4 (2011), no. 1, 60-69

##### AMA Style

JAVADIAN A., ESHAGHI GORDJI M., BAVAND SAVADKOUHI M., APPROXIMATELY PARTIAL TERNARY QUADRATIC DERIVATIONS ON BANACH TERNARY ALGEBRAS. J. Nonlinear Sci. Appl. (2011); 4(1):60-69

##### Chicago/Turabian Style

JAVADIAN, A., ESHAGHI GORDJI , M., BAVAND SAVADKOUHI, M.. " APPROXIMATELY PARTIAL TERNARY QUADRATIC DERIVATIONS ON BANACH TERNARY ALGEBRAS." Journal of Nonlinear Sciences and Applications, 4, no. 1 (2011): 60-69

### Keywords

• Hyers-Ulam-Rassias stability
• Banach ternary algebra

•  46K05
•  39B82
•  39B52
•  47B47

### References

• [1] S. Abbaszadeh, Intuitionistic fuzzy stability of a quadratic and quartic functional equation, Int. J. Nonlinear Anal. Appl. , 1 (2010), 100-124.

• [2] R. Badora, On approximate derivations, Math. Inequal. Appl. , 9 (2006), 167-173.

• [3] M. Bavand Savadkouhi, M. Eshaghi Gordji, J. M. Rassias, N. Ghobadipour, Approximate ternary Jordan derivations on Banach ternary algebras, J. Math. Phys, 50 (2009), 9 pages.

• [4] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. , 27 (1984), 76-86.

• [5] H. Chu, S. Koo, J. Park , Partial stabilities and partial derivations of n-variable functions, Nonlinear Anal.-TMA , (to appear),

• [6] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg , 62 (1992), 59-64.

• [7] A. Ebadian, A. Najati, M. E. Gordji, On approximate additive-quartic and quadratic- cubic functional equations in two variables on abelian groups, Results Math. , DOI 10.1007/s00025-010-0018-4 (2010)

• [8] M. Eshaghi Gordji , Stability of a functional equation deriving from quartic and additive functions, Bull. Korean Math. Soc. , 47 (2010), 491-502.

• [9] M. Eshaghi Gordji, Stability of an additive-quadratic functional equation of two variables in F-spaces, Journal of Nonlinear Sciences and Applications, 2 (2009), 251-259.

• [10] M. Eshaghi Gordji, S. Abbaszadeh, C. Park , On the stability of generalized mixed type quadratic and quartic functional equation in quasi-Banach spaces , J. Ineq. Appl., Article ID 153084, (2009), 26 pages.

• [11] M. Eshaghi Gordji, M. Bavand Savadkouhi, C. Park, Quadratic-quartic functional equations in RN-spaces , J. Ineq. Appl. Article ID 868423, (2009), 14 pages.

• [12] M. Eshaghi Gordji, M. Bavand Savadkouhi, M. Bidkham, Stability of a mixed type additive and quadratic functional equation in non-Archimedean spaces, Journal of Computational Analysis and Applications, 12 (2010), 454-462.

• [13] M. Eshaghi Gordji, A. Bodaghi, On the Hyers-Ulam-Rasias Stability problem for quadratic functional equations , East Journal On Approximations, 16 (2010), 123-130.

• [14] M. Eshaghi Gordji, A. Bodaghi, On the stability of quadratic double centralizers on Banach algebras, J. Comput. Anal. Appl., (in press),

• [15] M. Eshaghi Gordji, M. B. Ghaemi, S. Kaboli Gharetapeh, S. Shams, A. Ebadian, On the stability of $J^*$-derivations, Journal of Geometry and Physics. , 60(3) (2010), 454-459.

• [16] M. Eshaghi Gordji, M. Ghanifard, H. Khodaei, C. Park, A fixed point approach to the random stability of a functional equation driving from quartic and quadratic mappings, Discrete Dynamics in Nature and Society, Article ID: 670542. (2010)

• [17] M. Eshaghi Gordji, N. Ghobadipour, Stability of ($\alpha,\beta,\psi$)-derivations on Lie $C^*$-algebras, To appear in International Journal of Geometric Methods in Modern Physics , (IJGMMP),

• [18] M. Eshaghi Gordji, S. Kaboli Gharetapeh, T. Karimi , E. Rashidi, M. Aghaei , Ternary Jordan derivations on $C^*$-ternary algebras, Journal of Computational Analysis and Applications, 12 (2010), 463-470.

• [19] M. Eshaghi Gordji, S. Kaboli-Gharetapeh, C. Park, S. Zolfaghri , Stability of an additive-cubic-quartic functional equation, Advances in Difference Equations. Article ID 395693, (2009), 20 pages.

• [20] M. Eshaghi Gordji, S. Kaboli Gharetapeh, J. M. Rassias, S. Zolfaghari, Solution and stability of a mixed type additive, quadratic and cubic functional equation, Advances in difference equations, Article ID 826130, doi:10.1155/2009/826130. , 2009 (2009), 17 pages

• [21] M. Eshaghi Gordji, H. Khodaei , On the Generalized Hyers-Ulam-Rassias Stability of Quadratic Functional Equations, Abs. Appl. Anal. Article ID 923476, (2009), 11 pages.

• [22] M. Eshaghi Gordji, H. Khodaei, Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces, Nonlinear Analysis-TMA , 71 (2009), 5629-5643.

• [23] M. Eshaghi Gordji, M. S. Moslehian, A trick for investigation of approximate derivations, Math. Commun. , 15 (2010), 99-105.

• [24] M. Eshaghi Gordji, M. Ramezani, A. Ebadian, C. Park, Quadratic double centralizers and quadratic multipliers, Advances in Difference Equations , (in press),

• [25] M. Eshaghi Gordji, J. M. Rassias, N. Ghobadipour, Generalized Hyers-Ulam stability of the generalized (n; k)-derivations , Abs. Appl. Anal., Article ID 437931, 2009 (2009), 8 pages.

• [26] R. Farokhzad, S. A. R. Hosseinioun, Perturbations of Jordan higher derivations in Banach ternary algebras: An alternative fixed point approach, Internat. J. Nonlinear Anal. Appl. , 1 (2010), 42-53.

• [27] Z. Gajda , On stability of additive mappings, Internat. J. Math. Sci. , 14 (1991), 431-434.

• [28] P. G·avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl, 184 (1994), 431-436.

• [29] A. Grabiec, The generalized Hyers-Ulam stability of a class of functional equations, Publ. Math. Debrecen. , 48 (1996), 217-235.

• [30] D. H. Hyers, G. Isac, Th. M. Rassias, Stability of functional equations in several variables, Birkhaer, Basel (1998)

• [31] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci, 27 (1941), 222-224.

• [32] G. Isac, Th. M. Rassias, On the Hyers-Ulam stability of $\psi$-additive mappings, J. Approx. Theory, 72 (1993), 131-137.

• [33] K. Jun, D. Park, Almost derivations on the Banach algebra $C^n[0, 1]$, Bull. Korean Math. Soc. , 33 (1996), 359-366.

• [34] Pl. Kannappan, Quadratic functional equation and inner product spaces, Results Math., 27 (1995), 368-372.

• [35] H. Khodaei, M. Kamyar , Fuzzy approximately additive mappings, Int. J. Nonlinear Anal. Appl. , 1 (2010), 44-53.

• [36] H. Khodaei, Th. M. Rassias, Approximately generalized additive functions in several variables, Int. J. Nonlinear Anal. Appl. , 1 (2010), 22-41.

• [37] C. Park, 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), 7 pages

• [38] C. Park, A. Najati , Generalized additive functional inequalities in Banach algebras, Int. J. Nonlinear Anal. Appl. , 1 (2010), 54-62.

• [39] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. , 72 (1978), 297-300.

• [40] F. Skof, Propriet locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano, 53 (1983), 113-129.

• [41] S. M. Ulam , Problems in modern mathematics, Chapter VI, science ed., Wiley, New York (1940)