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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 $$ f(x+2y)+f(y+2z)+f(z+2x)-2f(x+y+z)=3f(x)+3f(y)+3f(z),\quad x,y,z\in X\eqno{(*)} $$ and prove that it is equivalent to the quadratic equation $$ f(x+y)=f(x-y)=2f(x)+2f(y),\quad x,y\in X. $$ In case $Y$ is a Banach space, the stability of equation ($*$) is proved. Namely, if $$\multline \|f(x+2y)+f(y+2z)+f(z+2x)-2f(x+y+z)-3f(x)-3f(y)-3f(z)\|\\ \leq\varphi(x,y,z),\quad x,y,z\in X \endmultline$$ with a suitable control function $\varphi: X^3\to [0,\infty)$, then there exists a unique solution $F: X\to Y$ of ($*$), close (in a specific sense) to $f$. In particular, if $\varphi(x,y,z)=\varepsilon$ ($\varepsilon\geq 0$), then $$ \|f(x)-F(x)\|\leq \frac{\varepsilon}{8},\quad x\in X. $$ Other considered forms of $\varphi$ are $\varphi(x,y,z)=\varepsilon(\|x\|^p+\|y\|^p+\|z\|^p)$ ($p\neq 2$) and $\varphi(x,y,z)=\varepsilon(\|x\|^{p_1}\|y\|^{p_2}\|z\|^{p_3})$ ($p_1+p_2+p_3\neq 2$).
39B82Stability, separation, extension, and related topics
39B52Functional equations for functions with more general domains and/or ranges
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