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.
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Brzdęk, J. On a generalization of the Cauchy functional equation. Aeq. Math. 46, 56–75 (1993). https://doi.org/10.1007/BF01833998
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DOI: https://doi.org/10.1007/BF01833998