On a generalization of the Cauchy functional equation

Summary

While looking for solutions of some functional equations and systems of functional equations introduced by S. Midura and their generalizations, we came across the problem of solving the equationg(ax + by) = Ag(x) + Bg(y) + L(x, y) (1) in the class of functions mapping a non-empty subsetP of a linear spaceX over a commutative fieldK, satisfying the conditionaP + bP ⊂ P, into a linear spaceY over a commutative fieldF, whereL: X × X → Y is biadditive,a, b ∈ K\{0}, andA, B ∈ F\{0}.

Theorem.Suppose that K is either R or C, F is of characteristic zero, there exist A ₁,A ₂,B ₁,B ₂,∈ F\ {0}with L(ax, y) = A ₁ L(x, y), L(x, ay) = A ₂ L(x, y), L(bx, y) = B ₁ L(x, y), and L(x, by) = B ₂ L(x, y) for x, y ∈ X, and P has a non-empty convex and algebraically open subset. Then the functional equation (1)has a solution in the class of functions g: P → Y iff the following two conditions hold: L(x, y) = L(y, x) for x, y ∈ X, (2)if L ≠ 0, then A ₁ =A ₂,B ₁ =B ₂,A = A ²₁ ,and B = B ²₁ . (3)

Furthermore, if conditions (2)and (3)are valid, then a function g: P → Y satisfies the equation (1)iff there exist a y ₀ ∈ Y and an additive function h: X → Y such that if A + B ≠ 1, then y ₀ = 0;h(ax) = Ah(x), h(bx) =Bh(x) for x ∈ X; g(x) = h(x) + y ₀ + 1/2A ^-1₁ B ^-1₁ L(x, x)for x ∈ P.