Let G, H be uniquely 2-divisible Abelian groups. We study the solutions f, g: G → H of Pexider type functional equation (*)$$f(x+y)+f(x-y)+g(x+y)=2f(x)+2f(y)+g(x)+g(y),$$f(x+y)+f(x−y)+g(x+y)=2f(x)+2f(y)+g(x)+g(y), resulting from summing up the well known quadratic functional equation and additive Cauchy functional equation side by side. We show that modulo a constant equation (*) forces f to be a quadratic function, and g to be an additive one (alienation phenomenon). Moreover, some stability result for equation (*) is also presented.

中文翻译：

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二次函数方程和可加函数方程的异化

令 G, H 是唯一可 2 整除的阿贝尔群。我们研究解 f, g: G → H 的 Pexider 型函数方程 (*)$$f(x+y)+f(xy)+g(x+y)=2f(x)+2f(y)+ g(x)+g(y),$$f(x+y)+f(x−y)+g(x+y)=2f(x)+2f(y)+g(x)+g( y)，由众所周知的二次函数方程和加性柯西函数方程并列相加得到。我们证明模常数方程 (*) 迫使 f 为二次函数，而 g 为可加函数（异化现象）。此外，还给出了方程（*）的一些稳定性结果。