1. Set up, notation and terminology

Throughout the paper we work in the following framework and with the following notation and terminology. We use it without explicit mentioning.

S is a *semigroup*
a set equipped with an associative composition rule (x,y) ↦ xy
, σ : S → S is an homomorphism satisfying σ ○ σ =id, and (H,+) denotes an abelian group which is *uniquely 2-divisible*, i.e., for any h ∈ H the equation 2x=h has exactly one solution x ∈ H.

A function a : S→ H is said to be *additive* if

(1.1) a(xy)=a(x)+a(y) for all x,y ∈ S.

A function 𝑓:S→ H is *abelian*, if

(x_{π(1)}x_{π(2)} ··· x_{π(k)})=𝑓(x_{1}x_{2}··· x_{k})

for all x_{1},x_{2},··· ,x_{k} ∈ S, all permutations π of k elements and all k=2,3, ···. Any abelian function 𝑓 is *central*, meaning (xy)=𝑓(yx) for all x,y ∈ S.

By 𝓝(S,H,σ) we mean the set of the solutions θ:S→ H of the homogeneous equation

θ(xy)-θ(σ(x)y)=0, x,y ∈ S.

We recall that the *Cauchy difference* C𝑓 of a function :S→ H is defined by

C𝑓(x,y):=𝑓(xy)-𝑓(x)-𝑓(y), x,y ∈ S.

If :S→ H is a function, then J_{𝑓}:S→ H and φ_{𝑓}: S→ H are defined by

2𝑓(x):=J_{𝑓}(x)+C𝑓(x,x) and φ_{𝑓}(x):=𝑓(σ(x)x)

for all x ∈ S.

2. Introduction

The purpose of the present paper is to solve the following *variant of the quadratic functional equation*

(2.1) (xy)+𝑓(σ(y)x)=2𝑓(x)+2𝑓(y), x,y ∈ S,

where 𝑓 :S→ H is the unknown function. The difference between (2.1) and the quadratic standard functional equation

(2.2) (xy)+𝑓(xσ(y))=2𝑓(x)+2𝑓(y), x,y ∈ S,

is that the new equation (2.1) has, on the second term, 𝑓(σ(y)x) while the old one (2.2) has 𝑓(xσ(y)). On abelian semigroups the functional Eqs. (2.1) and (2.2) coincide, and their solutions are known see e.g. (^{10}) and ((^{13}),Chapter 13), so the contributions of the present paper to the theory of quadratic functional equations lie in the non-abelian case.

A special case of (2.1) is the *symmetrized additive Cauchy equation*

(2.3) (xy)+𝑓(yx)=2𝑓(x)+2𝑓(y), x,y ∈ S.

Eq. (2.3) is a non-commutative version of the additive equation (1.1), because it reduces to (1.1) if S is abelian. On groups the solutions of (2.3) are according to ((^{13}), Proposition 2.17) the same as those solutions 𝑓 of *Jensen's functional equation* (xy)+𝑓(xy^{-1})=2𝑓(x) for which 𝑓(e)=0. Example 12.4 in (^{13}) present a non-abelian solution of (2.3) on the 3-dimensional Heisenberg group H_{3}(**R**). Therefore the functional equation (2.1) has in general non-abelian solutions.

Similar functional equations that have also been studied are

(2.4) (xy)+𝑓(σ(y)x) = 2𝑓(x)𝑓(y), x,y ∈ S,

(2.5) (xy)+𝑓(σ(y)x) = 2𝑓(x)g(y), x,y ∈ S,

(2.6) (xy)+𝑓(σ(y)x) = 2𝑓(x), x,y ∈ S.

The complex-valued solutions of (2.4) were determined by Stetkær in (^{14}), while the complex-valued solutions (𝑓,g), where S is a possibly non-abelian group or monoid, of (2.5) and the solutions 𝑓 :S→ H of (2.6) were obtained by the authors in (^{3}) and (^{4}), respectively. It turns out that, like on abelian groups, only multiplicative and additive functions occur in the solution formulas of (2.4), (2.5) and (2.6). We will prove that these contrast the solutions of the functional equation (2.1), where the non-abelian phenomena like solutions of (2.3) may occur. For other similar functional equations we refer to (^{1}),(^{2}),(^{5}),(^{7})-(^{9}),(^{11}),(^{12}),(^{16}).

One of the main results is that the solutions for the variant (2.1) of the quadratic functional equation

can be expressed in terms of bi-additive maps and solutions of the symmetrized additive Cauchy equation (Theorem 5.4), so that the form of the solutions generalizes the case where S is abelian, see e.g. ((^{10}),Theorem 3) and ((^{13}), Theorem 13.6). Thus the contribution by our paper of new knowledge is an extension of earlier results from the abelian to the non-abelian case because (2.3) becomes (1.1) if S is abelian.

As applications, two important results (Corollaries 5.3 and 5.6 about Drygas' type equation

(xy)+𝑓(σ(y)x)=2𝑓(x)+𝑓(y)+𝑓(σ(y)), x,y ∈ S,

are presented. Our solution formulas contain the abelian ones as special cases.

Finally, we note that the results about Whitehead's functional equation

\begin{equation}\label{eq27}

𝑓(xyz)=𝑓(xy)+𝑓(xz)+𝑓(yz)-𝑓(x)-𝑓(y)-𝑓(z), x,y,z ∈ S,

(2.7)

given in (^{15}) play an important role in finding solutions to the functional equation (2.1).

3. Results about Whitehead's functional equation (2.7)

The following lemma lists pertinent basic properties of any solution :S→ H of (2.7). For the notation J_{𝑓}, see Section 1.

**Theorem 3.1.** Let :S → H be a solution of (2.7). In that case

(a) C𝑓:S × S→ H is bi-additive.

(b) J_{𝑓}:S → H satisfies (2.3).

(c) If 𝑓 is central, then J_{𝑓} is additive.

(d) Let s ∈ Hom(S,S). If 𝑓 ○ s=𝑓, then J_{𝑓} ○ s=J_{𝑓}.

**Proof.** (a) and (b) can be found in (^{15})

(c) Let 𝑓 be central. To get that J_{𝑓} is central it suffices to prove that so is x↦ C𝑓(x,x). That is an easy task. The rest follows from (b).

(d) By the definition of J_{𝑓} it suffices to prove that C𝑓(s(x),s(x))=C𝑓(x,x) for all x ∈ S, and that follows from the definition of C𝑓.

4. Connections between (2.1) and (2.7)

Lemma 4.1 below derives one connection between (2.1) and (2.7), viz.,

**Lemma 4.1.** If :S → H satisfies (2.1), then it also satisfies (2.7).

**Proof.** Making the substitutions (xy,z), (σ(z)x,y), and (x,yz) in (2.1), we get respectively

(xyz)+𝑓(σ(z)xy) = 2𝑓(xy)+2𝑓(z),

𝑓(σ(z)xy))+𝑓(σ(yz)x) = 2𝑓(σ(z)x)+2𝑓(y),

(xyz)+𝑓(σ(yz)x) = 2𝑓(x)+2𝑓(yz).

Subtracting the middle identity from the sum of the other two we find that

2𝑓(xyz)=2𝑓(xy)+2𝑓(yz)+2𝑓(x)+2𝑓(z)-2𝑓(σ(z)x)-2𝑓(y).

Replacing here (σ(z)x) by 2𝑓(x)+2𝑓(z)-𝑓(xz) and using the fact that H is *uniquely 2-divisible*, we get (2.7).

In the following lemma, we derive another connection between (2.1) and (2.7).

**Lemma 4.1.** If :S → H satisfies (2.1), then φ_{𝑓}=J_{𝑓}.

**Proof.** The proof is a small computation, based on (2.1).

5. Results about (2.1)

We start with Lemma 5.1, in which we derive some properties of the solutions of (2.1).

**Lemma 5.1.** If :S → H satisfies (2.1), then

(a) 𝑓 ○ σ=𝑓.

(b) C𝑓(x,σ(y))=-C𝑓(y,x) for all x,y ∈ S.

(c) φ_{𝑓} is a solution of (2.1).

(d) φ_{𝑓} ∈ 𝓝 (S,H,σ), and φ_{𝑓} ○ σ = φ_{𝑓}.

**Proof:** (a) Let x,y ∈ S be arbitrary. Using (2.7) and (2.1), we obtain

On the other hand, we have

So 𝑓(y)-(σ(y))=0 for all y ∈ S, i.e., 𝑓 ○ σ = 𝑓.

(b) Let x,y ∈ S be arbitrary. By help of (a) and (2.1), we get that

C𝑓(x,σ(y)) = 𝑓(xσ(y))-𝑓(x)-𝑓(σ(y))

= (σ(x)y)-𝑓(x)-𝑓(y)

= 2(y)+2𝑓(x)-𝑓(yx)-𝑓(x)-𝑓(y)

= (y)+𝑓(x)-𝑓(yx)

= -C(y,x).

In (c) and (d) we use a couple of times that φ_{𝑓}=J_{𝑓} (Lemma 4.2).

(c) Recalling the definition of J_{𝑓} (Section 1) we get that 2𝑓(x)=C𝑓(x,x)+ φ_{𝑓}(x), so by linearity it suffices to show that the function x↦ C𝑓(x,x) is a solution of (2.1). And that is a simple computation, based on the bi-additivity of C𝑓 and (b).

(d) We read from Theorem 3.1(b) that φ(xy)+φ_{𝑓}(yx)=2φ_{𝑓}(x)+2φ_{𝑓}(y) for all x,y ∈ S. Comparing this with (c) gives the first statement of (d). That φ_{𝑓} ○ σ = φ_{𝑓} is a special instance of Theorem 3.1(d).

In the following theorem, we determine the central solutions :S → H of the functional equation (2.1). For abelian case it generalizes many results (see, e.g., ((^{10}),Theorem 3) and ((^{13}), Theorem 13.6)).

**Theorem 5.2.** The central solutions :S→ H of (2.1) are the functions of the form

𝑓(x)=Q(x,x)+a(x),

where Q : S× S → H is an arbitrary symmetric, bi-additive map such that Q(x,σ (y))=-Q(x,y) for all x,y ∈ S, and where a : S→ H is an arbitrary additive map such that a ○ σ = a.

**Proof.** Assume that :S → H is a central solution of (2.1). Since 𝑓 is central, then C𝑓 is symmetric and φ_{𝑓} is additive (Theorem 3.1 (c)). So 𝑓 has the desired form by the decomposition 2𝑓(x)=C𝑓(x,x)+φ_{𝑓}(x). Take Q=
C𝑓 and a =
φ_{𝑓}.

The other direction of the proof is trivial to verify.

As a consequence of Theorem 5.2, we have the following result on the central solutions of the functional equation

(5.1) (xy)+𝑓(σ (y)x)=2𝑓(x)+𝑓(y)+𝑓(σ (y)), x,y ∈ S,

which reveals a connection between (5.1) and (2.1) and contains the solution of Drygas' equation on commutative semigroups.

**Corollary 5.3.** The central solutions :S → H of (5.1) are the functions of the form

(5.2) 𝑓(x)=Q(x,x)+a(x),

where Q : S× S → H is an arbitrary symmetric, bi-additive map such that Q(x,σ(y))=-Q(x,y) for all x,y ∈ S, and where a: S → H is an arbitrary additive map.

**Proof.** It is easy to check that any function 𝑓 of the form (5.2) is central and satisfies (5.1). Conversely, assume that 𝑓 is a central solution of (5.1). Let 𝑓 _{e} and 𝑓_{o} denote the σ-even and the σ-odd parts of 𝑓, i.e.,

Simple computations show that 𝑓_{e} is a central solution of (5.1). Hence 𝑓_{e} is a central solution of (2.1). From Theorem 5.2, we see that there exist a symmetric, bi-additive map Q:S × S→ H with Q(x, σ (y))=-Q(x,y) for all x,y ∈ S, and an additive map a_{1}: S→ H with a_{1} ○ σ =a_{1} such that

𝑓_{e}(x)=Q(x,x)+a_{1}(x), x ∈ S.

On the other hand, since 𝑓_{o} = 𝑓-𝑓_{e}, 𝑓_{o} is also a solution of (5.1), so that 𝑓_{o} is a solution of the variant (2.6) of Jensen's functional equation. According to ((^{4})Theorem 3.2), we see that 𝑓_{o} is additive. Therefore 𝑓 = 𝑓_{e} + 𝑓_{o} has the required form with a=a_{1} + 𝑓_{o} and this completes the proof.

Now we treat the general case where the solution : S→ H need not be central.

**Theorem 5.4.** The general solution :S→ H of (2.1) is

(5.3) 𝑓(x)=Q(x,x) + ψ(x), x ∈ S,

where Q:S× S → H is an arbitrary bi-additive map such that Q(x,σ(y))=-Q(y,x) for all x,y ∈ S, and where ψ : S → H is an arbitrary solution of the symmetrized additive Cauchy equation such that ψ ○ σ= ψ and ψ ∈ 𝓝(S,H,σ).

**Proof.** Let :S→ H be a solution of (2.1). Using the decomposition 2𝑓(x)=C𝑓(x,x)+φ_{𝑓}(x), Theorem 3.1 and Lemma 5.1, we see that 𝑓 has the desired form. Take Q=
C𝑓 and ψ=
φ_{𝑓}.

The other direction of the proof is trivial to verify.

**Remark 5.1.** Theorem 5.4 is a non-abelian version of e.g. ((^{10}), Theorem 3) and ((^{13}), Theorem 13.6). Indeed, if S is abelian, then any solution of the symmetrized additive Cauchy equation reduces to an additive function and so the condition ψ ∈ 𝓝(S,H,σ) becomes ψ = ψ ○ σ.

In view of Theorem 5.4, we obtain the following result about solution, that need not be central, of Drygas' type equation (5.1) on any semigroup.

**Corollary 5.2.** The general solution :S→ H of (5.1) is

𝑓(x)=Q(x,x)+ψ (x)+a(x), x ∈ S,

where Q : S × S→ H is an arbitrary bi-additive map such that Q(x,σ(y))=-Q(y,x) for all x,y ∈ S, ψ :S → H is an arbitrary solution of the symmetrized additive Cauchy equation such that ψ ○ σ=ψ and ψ ∈ 𝓝(S,H,σ), and where a : S→ H is an additive function such that a ○ σ=-a.

**Proof.** As the proof of Corollary 5.3 with the necessary changes.