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The modified Hyers-Ulam-Rassias stability of a cubic type functional equation. (English) Zbl 1087.39027

The functional equation

$\begin{array}{c}f\left(x+y+2z\right)+f\left(x+y-2z\right)+f\left(2x\right)+f\left(2y\right)+7f\left(x\right)+7f\left(-x\right)\hfill \\ \hfill =2\left(f\left(x+y\right)+2f\left(x+z\right)+2f\left(x-z\right)+2f\left(y+z\right)+2f\left(y-z\right)\right)\phantom{\rule{2.em}{0ex}}\left(1\right)\end{array}$

of a cubic type (fulfilled e.g. by $f\left(x\right)=a{x}^{3}+b$) is considered for functions mapping a real vector space $X$ into a Banach space $Y$. Its general solution is given and the stability in the sense of Hyers, Ulam, Rassias and Găvruta is proved. Instead of the classical method of the “Hyers sequence” the so-called fixed point alternative is used in the proof. The desired cubic function near the approximate solution of (1) is the fixed point of some operator acting on functions $g:X\to Y$ such that $g\left(0\right)=0$.

##### MSC:
 39B82 Stability, separation, extension, and related topics 39B52 Functional equations for functions with more general domains and/or ranges