Solution of some irregular functional equations and their stability

In this note, we study the following functional equations: L(L(p, r) + L(q, r) + p + q, r) + L(L(p, r) + p, r) + L(q, r) = 0, L(L(p, r) + p + q + e, r) + L(p, r) = L(p + q, r) + pL(q, r), and L(p, q) = L(ζp, q), |ζ| 1, without any regularity assumption for all p, q, r ∈ A, where L : A^2 → A is defined by L(p, q) := g(p+q)−g(p)−g(q) for all p, q ∈ A. Also, we find general solutions of the above functional equations on algebras, unital algebras and real numbers, respectively. Finally, we investigate the stability of those functional equations in algebras and unital algebras, respectively.