Stability and superstability of ternary homomorphisms and ternary derivations on ternary quasi-Banach algebras

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In this article, we investigate the generalized Hyers-Ulam-Rassias stability, Isac-Rassias type stability and superstability of ternary homomorphisms and ternary derivations associated to the generalized m- variables Cauchy-Jensen functional equation ∑ i = 1 m f ( x i ) - 1 2 m ∑ i = 1 m f m x i + ∑ j = 1 , j ≠ i m x j + f ∑ i = 1 m x i = 0 for a fixed positive integer m with m ≥ 3 on ternary quasi-Banach algebras. 2010 Mathematics Subject Classification : 39B82; 39B52.

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Published 01 January 2012
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Osboueiet al.Advances in Difference Equations2012,2012:80 http://www.advancesindifferenceequations.com/content/2012/1/80
R E S E A R C HOpen Access Stability and superstability of ternary homomorphisms and ternary derivations on ternary quasiBanach algebras 1 2*3 45 Mostafa Osbouei , Madjid Eshaghi Gordji, Ali Ebadian , Gholamreza Asgariand Hassan Azadi Kenary
* Correspondence: madjid. eshaghi@gmail.com 2 Department of Mathematics, Semnan University, Semnan, Iran Full list of author information is available at the end of the article
Abstract In this article, we investigate the generalized HyersUlamRassias stability, IsacRassias type stability and superstability of ternary homomorphisms and ternary derivations associated to the generalizedmvariables CauchyJensen functional equation      m mm m    1    f(xi)f mxi+xj+f xi= 0 2m i=1i=1j=1,j=i i=1
for a fixed positive integermwithm3 on ternary quasiBanach algebras. 2010 Mathematics Subject Classification: 39B82; 39B52. Keywords:superstability, HyersUlamRassias stability, IsacRassiastype stability, tern ary algebra, homomorphism, derivation, quasiBanach space
1. Introduction A functional equation (ξ) is stable if any functiongsatisfying the equation (ξ) approxi mately is near to a true solution of (ξ). A functional equation (ξ) is superstable if any functiongsatisfying the equation (ξ) approximately is a true solution of (ξ). It is of interest to consider the concept of stability for a functional equation arising when we replace the functional equation by an inequality which acts as a perturbation of the equation. The first stability problem was raised by Ulam [1] during his talk at the University of Wisconsin in 1940. The stability question of functional equations is that how do the solutions of the inequality differ from those of the given functional equation? If the answer is affirmative, we would say that the equation is stable. In 1941, Hyers [2] gave a first affirmative answer to the question of Ulam for Banach spaces. Letf:E®Ebe a mapping between Banach spaces such that   f(x+y)f(x)f(y)δ
for allx, yÎE, and for someδ> 0. Then there exists a unique additive mappingT : E®Esuch that   f(x)T(x)δ
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