ON THE GENERALIZED HYERS-ULAM-RASSIAS STABILITY OF FUNCTIONAL EQUATIONS

Hassan Azadi Kenary, Yeol Je Cho

Abstract


Recently, in [4], Gordji and Khodaei proved the generalizedHyers-Ulam-Rassias stability for the quadratic functional equation\begin{eqnarray}\nonumberf(mx+ny)+f(mx-ny) &=&\frac{n(m+n)}{2}\big(f(x+y)+f(x-y)\big)\\&&\, +2(m^2-mn-n^2)f(x) +(n^2 -mn)f(y)\nonumber\end{eqnarray}for any fixed $m,n\in \mathbb{Z}$ with $n\neq \pm m, -3m$ and Koh, in [15], proved the generalizedHyers-Ulam-Rassias stability of cubic functional equation\begin{eqnarray}\nonumber 4f(x+my)+4f(x-my)+m^2 f(2x)=8f(x)+4m^2\big(f(x+y)+f(x-y)\big)\end{eqnarray}  for any fixed $m\in \mathbb{N}$ with $m\geq 2$  in Banach spaces.
In this paper, motivated by the results mentioned above, weprove the stability of these two functional equations and anotherone in random and non-Archimedean normed spaces.

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