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### Proyecciones (Antofagasta) vol.37 no.1 Antofagasta Mar. 2018

#### http://dx.doi.org/10.4067/S0716-09172018000100045

Articles

A variant of the quadratic functional equation on semigroups

1IBN TOFAIL University, Faculty of Sciences, Department of Mathematics, Kenitra, Morocco. E-mail : himfadli@gmail.com

2Moulay ISMAIL University, E. N. S. A. M, Department of Mathematics, Meknes, Morocco. E-mail : zeglamidriss@yahoo.fr

3IBN TOFAIL University, Faculty of Sciences, Department of Mathematics, Kenitra, Morocco. E-mail : samkabbaj@yahoo.fr

Abstract:

Let S be a semigroup, let H be an abelian group which is uniquely 2-divisible, and let σ be an involutive automorphism of S. We express the solutions : S → H of the following variant of the quadratic functional equation

(xy)+𝑓(σ(y)x)=2𝑓(x)+2𝑓(y), x,y ∈ S,

in terms of bi-additive maps and solutions of the symmetrized additive Cauchy equation.

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

(xπ(1)xπ(2) ··· xπ(k))=𝑓(x1x2··· xk)

for all x1,x2,··· ,xk ∈ 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 J𝑓:S→ H and φ𝑓: S→ H are defined by

2𝑓(x):=J𝑓(x)+C𝑓(x,x) and φ𝑓(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 H3(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.

\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).

The following lemma lists pertinent basic properties of any solution :S→ H of (2.7). For the notation J𝑓, 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) J𝑓:S → H satisfies (2.3).

(c) If 𝑓 is central, then J𝑓 is additive.

(d) Let s ∈ Hom(S,S). If 𝑓 ○ s=𝑓, then J𝑓 ○ s=J𝑓.

Proof. (a) and (b) can be found in (15)

(c) Let 𝑓 be central. To get that J𝑓 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 J𝑓 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 φ𝑓=J𝑓.

Proof. The proof is a small computation, based on (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) φ𝑓 is a solution of (2.1).

(d) φ𝑓 ∈ 𝓝 (S,H,σ), and φ𝑓 ○ σ = φ𝑓.

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 φ𝑓=J𝑓 (Lemma 4.2).

(c) Recalling the definition of J𝑓 (Section 1) we get that 2𝑓(x)=C𝑓(x,x)+ φ𝑓(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)+φ𝑓(yx)=2φ𝑓(x)+2φ𝑓(y) for all x,y ∈ S. Comparing this with (c) gives the first statement of (d). That φ𝑓 ○ σ = φ𝑓 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 φ𝑓 is additive (Theorem 3.1 (c)). So 𝑓 has the desired form by the decomposition 2𝑓(x)=C𝑓(x,x)+φ𝑓(x). Take Q= C𝑓 and a = φ𝑓.

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 𝑓o denote the σ-even and the σ-odd parts of 𝑓, i.e.,

Simple computations show that 𝑓e is a central solution of (5.1). Hence 𝑓e 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 a1: S→ H with a1 ○ σ =a1 such that

𝑓e(x)=Q(x,x)+a1(x), x ∈ S.

On the other hand, since 𝑓o = 𝑓-𝑓e, 𝑓o is also a solution of (5.1), so that 𝑓o is a solution of the variant (2.6) of Jensen's functional equation. According to ((4)Theorem 3.2), we see that 𝑓o is additive. Therefore 𝑓 = 𝑓e + 𝑓o has the required form with a=a1 + 𝑓o 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)+φ𝑓(x), Theorem 3.1 and Lemma 5.1, we see that 𝑓 has the desired form. Take Q= C𝑓 and ψ= φ𝑓.

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.

Acknowledgement

We wish to express our thanks to the referees for useful comments.

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Received: January 2017; Accepted: April 2017