Generalized Hyers-Ulam stabilities of an Euler-Lagrange-Rassias quadratic functional equation (Q2911882)

Let \(X,Y\) be real vector spaces, \(f: X\to Y\). The authors consider the functional equation NEWLINE\[NEWLINE f(x+2y)+f(y+2z)+f(z+2x)-2f(x+y+z)=3f(x)+3f(y)+3f(z),\quad x,y,z\in X\eqno{(*)} NEWLINE\]NEWLINE and prove that it is equivalent to the quadratic equation NEWLINE\[NEWLINE f(x+y)=f(x-y)=2f(x)+2f(y),\quad x,y\in X. NEWLINE\]NEWLINE In case \(Y\) is a Banach space, the stability of equation (\(*\)) is proved. Namely, if NEWLINE\[NEWLINE\begin{multlined} \|f(x+2y)+f(y+2z)+f(z+2x)-2f(x+y+z)-3f(x)-3f(y)-3f(z)\|\\ \leq\varphi(x,y,z),\quad x,y,z\in X \end{multlined}NEWLINE\]NEWLINE 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 NEWLINE\[NEWLINE \|f(x)-F(x)\|\leq \frac{\varepsilon}{8},\quad x\in X. NEWLINE\]NEWLINE Other considered forms of \(\varphi\) are \(\varphi(x,y,z)=\varepsilon(\|x\|^p+\|y\|^p+\|z\|^p)\) (\(p\neq 2\)) and \(\varphi(x,y,z)=\varepsilon(\|x\|^{p_1}\|y\|^{p_2}\|z\|^{p_3})\) (\(p_1+p_2+p_3\neq 2\)).