Results Math 72 (2017), 555–571
2017 Springer International Publishing
published online February 28, 2017
Results in Mathematics
A Variant of a Generalized Quadratic
Functional Equation on Groups
Heather Hunt Elfen, Thomas Riedel
, and Prasanna K. Sahoo
Abstract. Let G be a group and C the ﬁeld of complex numbers. Suppose
σ : G → G is an involutive endomorphism, that is, σ is an endomorphism
of G and it satisﬁes the condition σ(σ(x)) = x for all x in G. In this paper,
we ﬁnd the solutions f, g, h, k : G → C of the equation f(xy)+g(σ(y)x)=
h(x)+k(y) for all x, y ∈ G assuming f and g to be central functions.
This equation is a variant of a generalized quadratic functional equation
on groups with an involutive endomorphism. As an application, using the
solutions of this equation, we ﬁnd the solutions f, g, h, k : G × G → C of
the equation f(pr, qs)+g(sp, rq)=h(p, q)+k(r, s) for all p, q, r, s ∈ G
assuming f and g to be central functions.
Mathematics Subject Classiﬁcation. Primary 39B52, 39B82.
Keywords. Bihomomorphism, group, homomorphism, involutive
endomorphism, quadratic functional equation, semigroup.
The functional equation
for all x, y ∈ G, where G is a group written multiplicatively and y
inverse of y, is known as the quadratic functional equation and it serves in cer-
tain abstract spaces as the deﬁnition of norm. It was studied by many authors
including Jensen [9,10], Jordan and von Neumann , Kurepa , Acz´el
and Vincze , Acz´el , Kannappan [12–14], and Yang . The functional