Document Type: Research Paper

Author

Department of Mathematics, Payame Noor University, P.O. Box 19395-3697 Tehran, Iran.

Abstract

The stability problem of the functional equation was conjectured by Ulam and was solved by Hyers in the case of additive mapping. Baker et al. investigated the superstability of the functional equation from a vector space to real numbers. In this paper, we exhibit the superstability of $m$-additive maps on complete non--Archimedean spaces via a fixed point method raised by Diaz and Margolis.

Keywords

Main Subjects

[1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64--66.

[2] R. Badora, On approximate derivations, Math. Inequal. Appl. 9 (2006), no. 1, 167--173.

[3] J. Baker, J. Lawrence and F. Zorzitto, The stability of the equation $f(x + y) =f(x)f(y)$, Proc. Am. Math. Soc. 74 (1979), 242-–246

[4] D.G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math, Soc. 57 (1951), 223--237.

[5] L. C$breve{a}$dariu and  V. Radu, On the stability of the Cauchy functional equation: A fixed point approach,
Grazer Math. Ber. {346} (2004), 43--52.

[6] J.B. Diaz and B. Margolis, A fixed point theorem of the alternative for contractions on the generalized complete metric space, Bull. Amer. Math. Soc. {126} (1968) 305--309.

[7] A. Ebadian, I. Nikoufar, Th. M. Rassias, and N. Ghobadipour, Stability of generalized derivations on Hilbert C*-modules associated to a Pexiderized Cauchy-Jensen type functional equation, Acta Mathematica Scientia, {32B}(3) (2012), 1226--1238.

[8] P. Gu{a}vruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,
J. Math. Anal. Appl. {184} (1994), 431--436.

[9] P. Gu{a}vruta, An answer to a question of J. M. Rassias concerning the stability of Cauchy equation,
in: Advances in Equations and Inequalities, Hardronic Math. Ser. (1999), 67--71.

[10] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. {27} (1941), 222--224.

[11] D.H. Hyers, G. Isac, and Th. M. Rassias, Stability of functional equations in several variables, Birkh"{a}user, Boston, 1998.

[12] D.H. Hyers and Th. M. Rassias, Approximate homomorphisms, Aequations Math., 44 (1992), 125-153.

[13] I. Nikoufar and Th.M. Rassias, Approximately  algebraic tensor products, to appear in Miskolc Mathematical Notes.

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

[15] J.M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. {46} (1982), 126--130.

[16] V. Radu, The fixed point alternative and the stability of functional equations, Sem. Fixed Point Theory {4}(1) (2003) 91-96.

[17] S. M. Ulam, A collection of mathematical problems, Interscience Tracts in Pure and Applied Mathematics, Interscience Publisher, New York, 1960.