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Generalized Hyers-Ulam stabilities of an Euler-Lagrange-Rassias quadratic functional equation. (English) Zbl 1251.39028

Let $X,Y$ be real vector spaces, $f:X\to Y$. The authors consider the functional equation

$f\left(x+2y\right)+f\left(y+2z\right)+f\left(z+2x\right)-2f\left(x+y+z\right)=3f\left(x\right)+3f\left(y\right)+3f\left(z\right),\phantom{\rule{1.em}{0ex}}x,y,z\in X\phantom{\rule{2.em}{0ex}}\left(*\right)$

and prove that it is equivalent to the quadratic equation

$f\left(x+y\right)=f\left(x-y\right)=2f\left(x\right)+2f\left(y\right),\phantom{\rule{1.em}{0ex}}x,y\in X·$

In case $Y$ is a Banach space, the stability of equation ($*$) is proved. Namely, if

$\begin{array}{c}\parallel f\left(x+2y\right)+f\left(y+2z\right)+f\left(z+2x\right)-2f\left(x+y+z\right)-3f\left(x\right)-3f\left(y\right)-3f\left(z\right)\parallel \hfill \\ \hfill \le \phi \left(x,y,z\right),\phantom{\rule{1.em}{0ex}}x,y,z\in X\end{array}$

with a suitable control function $\phi :{X}^{3}\to \left[0,\infty \right)$, then there exists a unique solution $F:X\to Y$ of ($*$), close (in a specific sense) to $f$. In particular, if $\phi \left(x,y,z\right)=\epsilon$ ($\epsilon \ge 0$), then

$\parallel f\left(x\right)-F\left(x\right)\parallel \le \frac{\epsilon }{8},\phantom{\rule{1.em}{0ex}}x\in X·$

Other considered forms of $\phi$ are $\phi \left(x,y,z\right)=\epsilon \left(\parallel x{\parallel }^{p}+\parallel y{\parallel }^{p}+\parallel z{\parallel }^{p}\right)$ ($p\ne 2$) and $\phi \left(x,y,z\right)=\epsilon \left(\parallel x{\parallel }^{{p}_{1}}\parallel y{\parallel }^{{p}_{2}}\parallel z{\parallel }^{{p}_{3}}\right)$ (${p}_{1}+{p}_{2}+{p}_{3}\ne 2$).

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