We deal with an alienation problem for an Euler–Lagrange type functional equation

$$\begin{aligned} f(\alpha x + \beta y) + f(\alpha x - \beta y) = 2\alpha ^2f(x) + 2\beta ^2f(y) \end{aligned}$$

assumed for fixed nonzero real numbers \(\alpha ,\beta ,\, 1 \ne \alpha ^2 \ne \beta ^2\), and the classic quadratic functional equation

$$\begin{aligned} g(x+y) + g(x-y) = 2g(x) + 2g(y). \end{aligned}$$

We were inspired by papers of Kim et al. (Abstract and applied analysis, vol. 2013, Hindawi Publishing Corporation, 2013) and Gordji and Khodaei (Abstract and applied analysis, vol. 2009, Hindawi Publishing Corporation, 2009), where the special case \(g = \gamma f\) was examined.

中文翻译：

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两个二次型函数方程的异化

我们处理欧拉-拉格朗日型函数方程的异化问题

$$\begin{aligned} f(\alpha x + \beta y) + f(\alpha x - \beta y) = 2\alpha ^2f(x) + 2\beta ^2f(y) \end{aligned }$$

假定为固定的非零实数\(\alpha ,\beta ,\, 1 \ne \alpha ^2 \ne \beta ^2\)和经典的二次函数方程

$$\begin{对齐} g(x+y) + g(xy) = 2g(x) + 2g(y)。\end{对齐}$$

我们受到了 Kim 等人论文的启发。（抽象和应用分析，第 2013 卷，Hindawi Publishing Corporation，2013）和 Gordji 和 Khodaei（抽象和应用分析，第 2009 卷，Hindawi Publishing Corporation，2009），其中特例\(g = \gamma f\)被检查了。