Document Type : Research Paper


1 Department of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.

2 Department of Mathematics and Computer Sciences, Hakim Sabzevari University, P.O. Box 397, Sabzevar, Iran.


In this paper, we investigate the Hyers-Ulam stability of the orthogonally  cubic equation and  Pexiderized cubic equation
in multi-normed spaces by the direct method and the fixed point method. Moreover, we prove the Hyers-Ulam stability of the  $2$-variables cubic  equation
 f(2x+y,2z+t)+f(2x-y,2z-t) =2f(x+y,z+t) +2f(x-y,z-t)+12f(x,z),
and orthogonally cubic type and $k$-cubic equation in multi-normed spaces. A counter example for non stability of the cubic equation is also discussed.


Main Subjects

[1] I.S. Chang, K.W. Jun, and Y.S. Jung, The modified Hyers-Ulam-Rassias stability of a cubic type functional equation, Math. Inequal. Appl. 8 (4) (2005), 675-683.
[2] I.S. Chang and Y.S. Jung, Stability for the functional equation of cubic type, J. Math. Anal. Appl. 334 (2007), 85-96.
[3]  H.G. Dales and M.E. Polyakov, Multi-normed spaces, arxiv:1112.5148v2, 2012.
[4] S. Gudder and D. Strawther, Orthogonally additive and orthogonally increasing functions on vector spaces, Pacific J. Math. 58 (1975), 427-436.
[5] D.H. Hyers, On the stability of the linear functional equation, Proc.Nat. Acad.Sci.,U.S.A.,27 (1941), 222-224.
[6] K.W. Jun and M.M. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl. 274 (2), (2002), 267-278.
[7] S.M. Jung, M.Th. Rassias, and C. Mortici, On a functional equation of trigonometric type, Applied Mathematics and Computation, 252(2015), 294-303.
[8] M.S. Moslehian, K. Nikodem, and  D. Popa, Asymptotic aspect of the quadratic functional equation in multi-normed spaces, J.  Math. Anal.  Appl.  355 (2009), 717-724.
[9] S. Ostadbashi and M. Solaimaninia, On Pexider difference for a Pexider cubic functional equation, Reports 18 (2016), 151-162.
[10] A.G. Pinsker, Sur une fonctionnelle dans l’espace de Hilbert, Comptes Rendus (Dokl.) de l’Académie des Sciences, URSS, 20, (1938), 411-414.
[11]  Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297-300.
[12] J. Ratz, On orthogonally additive mappings, Aequationes Math. 28 (1985), 35-49.
[13] K. Ravi, M.J. Rassias, P. Narasimman, and R. K. Kumar, Stabilities of a general $k$-cubic functional equation in Banach spaces, Contemporary Anal.  Appl. Math. 3  (2015), 1-12.
[14]  S.M. Ulam, Problems in Modern Mathematics, Science Editions, Wiley, NewYork, 1964.
[15] T.Z. Xu, J. M. Rassias, and W.X. Xu, Generalized Hyers-Ulam stability of a general mixed additive-cubic functional equation in quasi-Banach spaces, Acta Math. Sinica, Eng. Ser. 28 (2012), 529-560.