Let \((S,+)\) be an abelian semigroup, let \((H,+)\) be a uniquely 2-divisible, abelian group and let \(\sigma ,\tau \) be two endomorphisms of S. In this paper, we find the solutions \(f:S\rightarrow H\) of the following Drygas type functional Eq.

$$\begin{aligned} f(x+\sigma (y))+f(x+\tau (y))=2f(x)+f(\sigma (y))+f(\tau (y)), \quad x,y\in S, \end{aligned}$$(1)

in terms of additive and bi-additive maps. Further, we solve a partly Pexiderized version of the above Eq and we use its solutions to determine the solutions of some related functional Eqs. In all results it is assumed that at least one of the endomorphisms \(\sigma \) and \(\tau \) is surjective.

中文翻译：

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具有内同态的Abelian半群上的Drygas函数方程

令\（（（S，+）\）为阿贝尔半群，令\（（H，+）\）为唯一可除以2的阿贝尔群，令\（\ sigma，\ tau \）为S的两个内同构。在本文中，我们找到以下Drygas型泛函方程的解\（f：S \ rightarrow H \）。

$$ \ begin {aligned} f（x + \ sigma（y））+ f（x + \ tau（y））= 2f（x）+ f（\ sigma（y））+ f（\ tau（y））， \ quad x，y \ in S，\ end {aligned} $$（1）

就加性图和双加性图而言。此外，我们解决了上述等式的部分受Pexiderized版本，并使用其解决方案来确定一些相关功能等式的解决方案。在所有结果中，假定内同形\（\ sigma \）和\（\ tau \）中的至少一个是射影。