Let X, Y be linear spaces over a field \({\mathbb {K}}\). Assume that \(f :X^2\rightarrow Y\) satisfies the general linear equation with respect to the first and with respect to the second variables, that is,

for all \(x,x_i,y,y_i \in X\) and with \(a_i,\,b_i \in {\mathbb {K}}{\setminus } \{0\}\), \(A_i,\,B_i \in {\mathbb {K}}\) (\(i \in \{1,2\}\)). It is easy to see that such a function satisfies the functional equation

for all \(x_i,y_i \in X\) (\(i \in \{1,2\}\)), where \(C_1:=A_1B_1\), \(C_2:=A_1B_2\), \(C_3:=A_2B_1\), \(C_4:=A_2B_2\). We describe the form of solutions and study relations between \((*)\) and \((**)\).

设X ,  Y是域\({\mathbb {K}}\) 上的线性空间。假设\(f :X^2\rightarrow Y\)满足关于第一个变量和关于第二个变量的一般线性方程，即，

对于所有\(x,x_i,y,y_i \in X\)和\(a_i,\,b_i \in {\mathbb {K}}{\setminus } \{0\}\) , \(A_i, \,B_i \in {\mathbb {K}}\) ( \(i \in \{1,2\}\) )。不难看出，这样的函数满足函数方程

对于所有\(x_i,y_i \in X\) ( \(i \in \{1,2\}\) )，其中\(C_1:=A_1B_1\) , \(C_2:=A_1B_2\) , \( C_3:=A_2B_1\) , \(C_4:=A_2B_2\)。我们描述了解决方案的形式并研究了\((*)\)和\((**)\) 之间的关系。