Asymptotic aspect of quadratic functional equations and super stability of higher derivations in multi-fuzzy normed spaces
Subject Areas : History and biographyM. khanehgir 1 , F. Hasanvand 2
1 - Department of Mathematics, Mashhad Branch, Islamic Azad University, P.O.Box
91735, Mashhad, Iran
2 - Department of Mathematics, Mashhad Branch, Islamic Azad University, P.O.Box
91735, Mashhad, Iran
Keywords: fuzzy normed space, higher derivation, Hyers-Ulam-Rassias stability, quadratic functional equation, multi-normed space,
Abstract :
In this paper, we introduce the notion of multi-fuzzynormed spaces and establish an asymptotic behavior of the quadraticfunctional equations in the setup of such spaces. We theninvestigate the superstability of strongly higher derivations in theframework of multi-fuzzy Banach algebras
[1] T. Bag, S. K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. (2003), 687-705.
[2] T. Bag, S. K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets Syst. 151 (2005), 513-547.
[3] S. C. Cheng, J. N. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Calcutta Math. Soc. 86 (5) (1994), 429-436.
[4] Y. J. Cho, C. Park and Y. Yang, Stability of derivations in fuzzy normed algebras, J. Nonlinear Sci. Appl. 8 (2015), 1-7.
[5] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76-86.
[6] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg, 62 (1992), 59-67.
[7] H. G. Dales, Banach algebras and automatic continuity, London: Math. Soc. Monographs, New Series, 24. Oxford University Press, Oxford, 2000.
[8] H. G. Dales, M. E. Polyakov, Multi-normed spaces and multi-Banach algebras, preprint, 2008.
[9] H. G. Dales, M. S. Moslehian, Stability of mappings on multi-normed spaces, Glasg. Math. J. 49 (2) (2007), 321-332.
[10] M. Eshaghi Gordji, N. Ghobadipour, A. Najati and A. Ebadian, Almost Jordan homomorphisms and Jordan derivations on fuzzy Banach algebras, Funct. Anal. Approx. Comput. 2 (2012), 1-7.
[11] C. Felbin, Finite-dimensional fuzzy normed linear spaces, Fuzzy Sets Syst. 48 (2) (1992), 239-248.
[12] H. Hasse, F. K. Schmidt, Noch eine Begr¨undung der Theorie der h¨oheren Differentialquotienten in einem algebraischen Funktionenk¨orper einer Unbestimmten (German) J. Reine Angew. Math. 177 (1937), 215-237.
[13] S. Hejazian, T. L. Shatery, Automatic continuity of higher derivations on JB∗-algebras, Bull. Iranian Math. Soc. 33 (1) (2007), 11-23.
[14] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224.
[15] K. W. Jun, Y. W. Lee, The image of a continuous strong higher derivation is contained in the radical, Bull. Korean Math. Soc. 33 (1996), 229-232.
[16] A. K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets Syst. 12 (2) (1984), 143-154.
[17] S. V. Krishna, K. K. M. Sarma, Separation of fuzzy normed linear spaces, Fuzzy Sets Syst. 63 (2) (1994), 207-217.
[18] J. B. Miller, Higher derivations on Banach algebras, Amer. J. Math. 92 (1970), 301-331.
[19] M. Mirzavaziri and M.S. Moslehian, Automatic continuity of σ- derivations in C∗-algebras, Proc. Amer. Math. Soc. 134 (11) (2006), 3319-3327.
[20] M. S. Moslehian, K. Nikodem and D. Popa, Asymptotic aspect of the quadratic functional equation in multinormed spaces, J. Math. Anal. Appl. 355 (2009), 717-724.
[21] M. S. Moslehian, H. M. Sirvastava, Jensen’s functional equation in multi-normed spaces, Taiwanese J. Math. 14 (2) (2010), 453-462.
[22] M. S. Moslehian, Superstability of higher derivations in multi-Banach algebras, Tamsui Oxford J. Math. Sci. 24 (4) (2008), 417–427.
[23] C. Park, Generalized Hyers-Ulam stability of quadratic functional equations: a fixed point approach, Fixed Point Theory Appl. (2008), Art. ID 493751, 1-9.
[24] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.
[25] T. L. Shatery, Superstability of generalized higher derivations, Abstr. Appl. Anal. (2011), Art. ID 239849, 1-9.
[26] F. Skof, Proprieta locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano. 53 (1983), 113-129. [27] S. M. Ulam, A collection of the mathematical problems, Interscience Publ. NewYork, 1960.