Document Type : Research Paper

Author

Department of Mathematics, Faculty of Science, Urmia University, P.O.Box 165, Urmia, Iran.

Abstract

In this paper, we investigate approximations of the $k-th$ partial ternary cubic derivations on non-Archimedean $\ell$-fuzzy Banach ternary algebras and non-Archimedean $\ell$-fuzzy $C^{*}$-ternary algebras. First, we study non-Archimedean and $\ell$-fuzzy spaces, and then  prove the stability of partial ternary cubic $*$-derivations on non-Archimedean $\ell$-fuzzy $C^{*}$-ternary algebras. We therefore provide a link among different disciplines: fuzzy set theory, lattice theory, non-Archimedean spaces, and mathematical analysis.

Keywords

[1] M.A. Abolfathi, A. Ebadian and R. Aghalary, Stability of mixed additive-quadratic Jensen type functional equation in non-Archimedean $ell$-fuzzy normed spaces, Ann. Univ. Ferrara Sez. VII Sci. Mat., 60(2) (2014), pp. 307-319.
[2] M. Amini and R. Saadati, Topics in fuzzy metric space, J. Fuzzy. Math., 4 (2003), pp. 765-768.
[3] T. Aoki, On the stability of linear trasformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), pp. 64-66.
[4] B. Arsalan and H. Inceboz, Nearly k-th Partial Ternary Quadratic $*$-Derivations, Kyungpook Math. J., 55 (2015), pp. 893-907.
[5] A. Cayley, On the 34 concomitants of the ternary cubic, Am. J. Math., 4 (1981), pp. 1-15.
[6] S.C. Cheng and J.N. Mordeson, Fuzzy linear operator and fuzzy normed linear spaces, Bull. Calcutta Math. Soc., 86 (1994), pp. 429-436.
[7] P. Czerwik, Functional Equations and Inequalities in Several Variable, World Scientific Publishing Company, New Jersey, Hong Kong, Singapore and London, 2002.
[8] G. Deschrijver, D. O'Regan, R. Saadati and S.M. Vaezpour, $ell$-fuzzy Euclidean normed spaces and compactness, Chaos Solitions Fractals, 42 (2009), pp. 40-45.
[9] A. Ebadian, R. Aghalary and M.A. Abolfathi, On approximate dectic mappings in non-Archimedean spaces: a fixed point approah, Int. J. Nonlinear Anal. Appl., 5(2) (2014), pp. 111-–122.
[10] A. Ebadian, N. Ghobadipour, B. Savadkouhi and M. Eshaghi Gordji, of a mixed type cubic and quartic functional equation in non-Archimedean $ell$-fuzzy normed spaces, Thai J. Math., 9 (2011), pp. 243-259.
[11] M. Eshaghi, M.B. Savadkouhi, M. Bidkham, C. park and J.R. Lee, Nearly partial derivations on Banach ternary algebras, J. Math. Stat., 6 (4) (2010) pp. 454-461.
[12] P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of the approximately additive mappings, J. Math. Anal. Appl., 184 (1994), pp. 431-436.
[13] A. George and p. Veeramani, On some result in fuzzy metric space, Fuzzy Sets Syst., 64 (1994), pp. 395-399.
[14] J.A. Goguen, L-fuzzy sets, J. Math. Anal. Appl., 18 (1967), pp. 145-174.
[15] F.Q. Gouvea, p-Adic Numbers. An Introduction, Springer-Verlag, Berlin, 1997.
[16] O. Hadzic and E. Pap, Fixed point Theory in Probabilistic Metric Spaces, Kluwer Academic, Dordrecht, 2001.
[17] O. Hadzic, E. Pap and M. Budincevic, Countable extension of triangular norms and their applications to the fixed point theory in probabilistic metric spaces, Kybernetica, 38 (2002), pp. 363-381.
[18] K. Hensel, Uber eine neue Begundung der Theorie der algebraischen Zahlen, Jahresbericht der Deutschen Mathematiker-Vereinigung, 6 (1897), pp. 83-88.
[19] A. Himbert, Comptes Rendus del'Acad. Sci., Paris, (1985).
[20] N.E. Hoseinzadeh, A. Bodaghi and M.R. Mardanbeigi, Almost Multi-Cubic Mappings and a Fixed point Application, Sahand Commun. Math. Anal., 17 (3) (2020), pp. 131-142.
[21] D.H. Hyers, On the Stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A., 27 (1941), pp. 222-224.
[22] D.H. Hyers, G. Isac and Th.M. Rssias, Stability of Functional Equation in Several Variables, Birkh"ause, Basel, 1998.
[23] K. Jun and H. Kim, The gegeralized Hyers-Ulam-Rassias stability of cubic functional equation, J. Math. Anal. Appl., 274 (2002), pp. 867-878.
[24] O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Set Syst., 12 (3) (1984), pp. 1-7.
[25] M. Kapranov, IM. Gelfand and A. Zelevinskii, Discrimininants, Reesultants and Multidimensional Determinants( Modern Bikhauser Classics), Berlin, (1994).
[26] A.K. Katsaras, Fuzzy topological vector spaces, Fuzzy Set Syst., 12 (1984), pp. 143-154.
[27] R. Kerner, The cubic chessboard: Geometry and physics, Class. Quantum Grav., 14 (1997), pp. A203-A225 .
[28] A. Khrennikov, Non-Archimedean Analysis: Quantu Paradoxes, Dynamical Systems and Biological Models, Math. Appl., vol.427, Kluwer Academic publisher Dordrecht, 1997.
[29] S.V. Krishna and K.K.M. Sarma, Separation of fuzzy normed linear spaces, Fuzzy Set Syst., 63 (1994) 207-217.
[30] A.K. Mirmostafaee and M.S. Moslehian, Stability of additive mapping in non Archimedean fuzzy normed spaces, Fuzzy Set Syst., 160 (2009), 1643-1652.
[31] A. Najati, B. Noori and M.B. Moghimi, On Approximation of Some Mixed Functional Equations, Sahand Commun. Math. Anal., 18 (1) (2021), pp. 35-46.
[32] M. Nazarianpoor and G. Sadeghi, On the stability of the Pexiderized cubic functionalequation in multi-normed spaces, Sahand Commun. Math. Anal., 9 (1) (2018), pp. 45-83.
[33] S. Okabo, Triple products and Yang-Baxter equation I, II. Octonionic and quaternionic triple systems, J. Math. Phys., 34(7) (1993), 3273-3291 and
3291-3315.
[34] K.H. Park and Y.S. Jung, Stability of a cubic functional equation on groups, Bull. Korean Math. Soc., 41 (2004) 347–357.
[35] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978) 297-300.
[36] Th.M. Rassias, Functional Equation, Inequalities and Applications, KLuwer Academic publishers Co., Dordrecht, Boston, London, 2003.
[37] A.M. Robert, A Course in p-Adic Analysis,(Graduate Texts in Mathematics), Vol.198, Springer-Verlag, New York, 2000.
[38] R. Saadati, On the $ell$-fuzzy topological spaces, Chaos Solitions Fractals, 37 (2008), pp. 1419-1426.
[39] R. Saadati and J. Park, On the intuitionistic fuzzy topological spaces, Chaos Solitions Fractals, 27 (2006), pp. 331-344.
[40] B. Schweizer and A. Sklar, Probabilistic Metric Spaces, Elsevier, North Holand, New York, 1983.
[41] S. Shakeri, R. Saadati and C. Park, Stability of the functional equations in non- Archimedean $ell$-fuzzy normed spaces, Int. J. Nonlinear Anal. Appl., 1(2) (2010), pp. 72-83.
[42] N. Shilkret, Non-Archimedian Banach algebras, Ph.D. thesis, Polytechnic University, 1968.
[43] S.M. Ulam, Problem in Modern Mathematics, Chapter VI, Science Editions, Wiley, New York, 1964.
[44] J.Z. Xiao and X.H. Zhu, Fuzzy normed spaces of operators and its completeness, Fuzzy Set Syst., 133 (2003) pp. 389-399.
[45] L.A. Zadeh, Fuzzy sets, Inf. Control, 8 (1965) pp. 338-353.