A Variant of a Generalized Quadratic Functional Equation on Groups

Yet another variant of the Drygas functional equation on groups

“…Then it reduces to (1.4) for all x, y ∈ G × G. Since f is central, from Theorem 5.1 with σ 1 = σ 2 = σ, we obtain f (x) = A(x) + B(x, x) + β, g(x) = A(x) + B(x, x) + β + α, h(x) = φ(x), k(x) = A(σ(x)) + A(x) + 2B(σ(x), σ(x)) − φ(σ(x)) + α, where B ∈ SBihom((G×G)×(G×G), C) satisfying B(σ(x), y) = −B(x, y) for all x, y ∈ G × G, A ∈ Hom(G × G, C), φ : G × G → C is any function, and α, β are complex constants. Since x ∈ G × G, we can write x = (p, q) for p, q ∈ G. Hence we have f (p, q) = A(p, q) + B((p, q), (p, q)) + β, g(p, q) = f (p, q) + α, h(p, q) = φ(p, q), k(p, q) = A(q, p) + A(p, q) + 2B((q, p), (q, p)) − φ(q, p) + α for all p, q ∈ G. Using Lemma 5.2 from [4], the functions A can be decomposed as A(p, q) = θ 1 (p) + θ 2 (q), where θ 1 , θ 2 ∈ Hom(G, C). Using Lemma 6.1, the bihommorphism B can be further simplified as B((p, q), (p, q)) = ψ(pq −1 , pq −1 ).…”

“…f (xy) + f (xσ(y)) = 2f (x) + 2f (y) (1.1) for all x, y ∈ S, where S is a commutative semigroup and σ : S → S is an endomorphism of S such that σ(σ(x)) = x for all x ∈ S. In [4], a variant of the quadratic functional equation, namely f (xy) + f (σ(y)x) = 2f (x) + 2f (y) (1.2) for all x, y ∈ G, where G is a group (not necessarily abelian) was considered. If G is an abelian group or f is a central function on group G, then the equations (1.1) and (1.2) are equivalent.…”