In this paper we present some methods for solving a large class of composite functional equations. These methods are then applied to the functional equations

$$\begin{aligned} f(a f(x) f(y) + b(f(x)y+f(y)x) +cxy)= & {} f(x) f(y) \end{aligned}$$(1)$$\begin{aligned} f(a f(x)^k y + b f(y)^\ell x + cxy)= & {} f(x) f(y) \end{aligned}$$(2)

with \(a, b, c \in {\mathbb {R}}\) and \(k, \ell \in {\mathbb {N}} \cup \{0\}\), for which we obtain all continuous solutions \(f: {\mathbb {R}} \rightarrow {\mathbb {R}}\). These equations generalize some well-known composite functional equations.

中文翻译：

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关于求解复合函数方程的方法

在本文中，我们提出了一些解决大型复合函数方程的方法。然后将这些方法应用于函数方程

$$ \ begin {aligned} f（af（x）f（y）+ b（f（x）y + f（y）x）+ cxy）=＆{} f（x）f（y）\ end {已对齐} $$（1）$$ \开始{已对齐} f（af（x）^ ky + bf（y）^ \ ell x + cxy）=＆{} f（x）f（y）\ end {aligned } $$（2）

与\（a，b，c \ in {\ mathbb {R}} \）和\（k，\ ell \ in {\ mathbb {N}} \ cup \ {0 \} \）中，我们可以获得所有连续解\（f：{\ mathbb {R}} \ rightarrow {\ mathbb {R}} \）。这些方程式概括了一些众所周知的复合函数方程式。