On a general bilinear functional equation

Abstract Let X , Y be linear spaces over a field $${\mathbb {K}}$$ K . Assume that $$f :X^2\rightarrow Y$$ f : X 2 → Y satisfies the general equation with respect to first and second variables, is, for all $$x,x_i,y,y_i \in X$$ x , i y ∈ $$a_i,\,b_i {\mathbb {K}}{\setminus } \{0\}$$ a b \ { 0 } $$A_i,\,B_i A B ( $$i \{1,2\}$$ 1 ). It is easy see such function functional $$x_i,y_i ), where $$C_1:=A_1B_1$$ C = $$C_2:=A_1B_2$$ $$C_3:=A_2B_1$$ 3 $$C_4:=A_2B_2$$ 4 We describe form of solutions study relations between $$(*)$$ ( ∗ ) $$(**)$$