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Ulam Stability Generalizations of 4th- Order Ternary Derivations Associated to a Jmrassias Quartic Functional Equation on Fréchet Algebras

Ebadian, Ali

  • Received : 2011.02.19
  • Accepted : 2011.11.24
  • Published : 2013.06.23

Abstract

Let $\mathcal{A}$ be a Banach ternary algebra over a scalar field R or C and $\mathcal{X}$ be a ternary Banach $\mathcal{A}$-module. A quartic mapping $D\;:\;(\mathcal{A},[\;]_{\mathcal{A}}){\rightarrow}(\mathcal{X},[\;]_{\mathcal{X}})$ is called a $4^{th}$- order ternary derivation if $D([x,y,z])=[D(x),y^4,z^4]+[x^4,D(y),z^4]+[x^4,y^4,D(z)]$ for all $x,y,z{\in}\mathcal{A}$. In this paper, we prove Ulam stability generalizations of $4^{th}$- order ternary derivations associated to the following JMRassias quartic functional equation on fr$\acute{e}$che algebras: $$f(kx+y)+f(kx-y)=k^2[f(x+y)+f(x-y)]+2k^2(k^2-1)f(x)-2(k^2-1)f(y)$$.

Keywords

Ulam stability;Quartic functional equation;Fr$\acute{e}$chet algebras;Ternary Banach algebras;$4^{th}$- order ternary derivation

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