On a general bilinear functional equation

Abstract

Let X, Y be linear spaces over a field K. Assume that f : X-2 -> Y satisfies the general linear equation with respect to the first and with respect to the second variables, that is, {f(a(1)x(1) + a(2)x(2), y) = A(1)f(x(1), y) + A(2)f(x(2), y) f(x, b(1)y(1) + b(2)y(2)) = B(1)f(x, y(1)) + B(2)f(x, y(2)), (*) for all x, x(i), y, y(i) is an element of X and with a(i), b(i) is an element of K\{0}, A(i), B-i is an element of K (i is an element of {1, 2}). It is easy to see that such a function satisfies the functional equation f(a(1)x(1) + a(2)x(2), b(1)y(1) + b(2)y(2)) = C(1)f(x(1), y(1)) + C(2)f(x(1), y(2)) + C(3)f(x(2), y(1)) + C(4)f(x(2), y(2)), (**) for all x(i), y(i) is an element of X (i is an element of {1, 2}), where C-1 := A(1)B(1), C-2 := A(1)B(2), C-3 := A(2)B(1), C-4 := A(2)B(2). We describe the form of solutions and study relations between (*) and (**).