A new type of functional equations on semigroups with involutions
aa r X i v : . [ m a t h . F A ] F e b A NEW TYPE OF FUNCTIONAL EQUATIONS ONSEMIGROUPS WITH INVOLUTIONS
IZ-IDDINE EL-FASSI ∗ Abstract.
Let S be a commutative semigroup, K a quadratically closedcommutative field of characteristic different from 2, G a 2-cancellative abeliangroup and H an abelian group uniquely divisible by 2. The aim of this pa-per is to determine the general solution f : S → K of the d’Alembert typeequation: f ( x + y, z + w ) + f ( x + σ ( y ) , z + τ ( w )) = 2 f ( x, z ) f ( y, w ) , ( x, y, z, w ∈ S )the general solution f : S → G of the Jensen type equation: f ( x + y, z + w ) + f ( x + σ ( y ) , z + τ ( w )) = 2 f ( x, z ) , ( x, y, z, w ∈ S )the general solution f : S → H of the quadratic type equation quation: f ( x + y, z + w )+ f ( x + σ ( y ) , z + τ ( w )) = 2 f ( x, z )+2 f ( y, w ) , ( x, y, z, w ∈ S )where σ, τ : S → S are two involutions. Notation and terminology
Throughout the paper we work in the following framework and with thefollowing notation and terminology.( S, +) is a commutative semigroup, K is a field of characteristic differentfrom 2, G is a 2-cancellative abelian group and H is an abelian group uniquelydivisible by 2.(i) A mapping σ : S → S is called involution if σ ( σ ( x )) = x and σ ( x + y ) = σ ( x ) + σ ( y ) , x, y ∈ S. (ii) We say that χ : S → K is a multiplication function if χ ( x + y, z + w ) = χ ( x, z ) χ ( y, w ) , x, y, z, w ∈ S. (iii) A function A : S → G is called additive if A ( x + y, z + w ) = A ( x, z ) + A ( y, w ) , x, y, z, w ∈ S. ∗ Corresponding author.2010
Mathematics Subject Classification.
Key words and phrases.
Functional equations, Involution, Semigroups. (iv) We say that B : S × S → H is a biadditive function if B ( u + v, w ) = B ( u, w ) + B ( v, w ) , u, v, w ∈ S , and B ( u, v + w ) = B ( u, v ) + B ( u, w ) , u, v, w ∈ S . Introduction
The functional equation f ( x + y ) + f ( x − y ) = 2 f ( x ) f ( y ) , x, y ∈ R (2.1)is known as the scalar d’Alembert functional equation. It has a long historygoing back to d’Alembert [8]. This functional equation has an obvious extensionto groups, and even to semigroups with an involution, and a satisfactory theoryof its solutions exists as described in [16, Chapter 9] (also, see [2,9,10] for detailsand references).In 1989, Acz´el and Dhombres [2] proved that a mapping Q : X → Y satisfiesthe quadratic functional equation Q ( x + y ) + Q ( x − y ) = 2 Q ( x ) + 2 Q ( y ) (2.2)if and only if there exists a symmetric bi-additive mapping b : X → Y suchthat Q ( x ) = b ( x, x ), where b ( x, y ) := 14 [ Q ( x + y ) + Q ( x − y )] , x, y ∈ X, where X and Y are two vector spaces. Later, many different quadratic func-tional equations were solved by numerous authors; for example, see [7, 12, 13].Let X and Y be real vector spaces. For a given involution σ : X → X, thefunctional equation g ( x + y ) + g ( x + σ ( y )) = 2 g ( x ) + 2 g ( y ) , (2.3)is called the quadratic functional equation with involution. According to [15,Corollary 8], a function g : X → Y is a solution of (2.3) if and only if there existan additive function a : X → Y, and a bi-additive symmetric function b : X → Y such that a ( σ ( x )) = a ( x ), b ( σ ( x ) , y ) = − b ( x, y ) and g ( x ) = a ( x ) + b ( x, x ) forall x, y ∈ X. In 2010, Sinopoulos [14] determined the general solution of the followingfunctional equations: g ( x + y ) + g ( x + σ ( y )) = 2 g ( x ) g ( y ) , x, y ∈ S, (2.4) g ( x + y ) + g ( x + σ ( y )) = 2 g ( x ) , x, y ∈ S, (2.5) NEW TYPE OF FUNCTIONAL EQUATIONS 3 g ( x + y ) + g ( x + σ ( y )) = 2 g ( x ) + 2 g ( y ) , x, y ∈ S, (2.6)where ( S, +) is a commutative semi-group and σ : S → S is an involution.The equation (2.5), in the case where S is an abelian group divisible by 2and σ ( x ) = − x , is equivalent to the Jensen equation J (cid:18) x + y (cid:19) = J ( x ) + J ( y )2 , (2.7)and is easily reduced to the Cauchy equation (see [3]).The equation (2.6), again with σ ( x ) = − x , plays a fundamental role in thecharacterization of inner product spaces [1, 3, 4, 11].The aim of this paper is to solve the following functional equations: f ( x + y, z + w ) + f ( x + σ ( y ) , z + τ ( w )) = 2 f ( x, z ) f ( y, w ) (2.8) f ( x + y, z + w ) + f ( x + σ ( y ) , z + τ ( w )) = 2 f ( x, z ) (2.9) f ( x + y, z + w ) + f ( x + σ ( y ) , z + τ ( w )) = 2 f ( x, z ) + 2 f ( y, w ) (2.10)for all x, y, z, w ∈ S, where ( S, +) is a commutative semi-group and σ, τ aretwo involutions.The functional equation f ( x + y, z + w ) = f ( x, z ) f ( y, w ) , x, y, z, w ∈ S (2.11)corresponds to σ ( x ) = τ ( x ) = x in (2.8), and the functional equation f ( x + y, z + w ) = f ( x, z ) + f ( y, w ) , x, y, z, w ∈ S (2.12)corresponds to σ ( x ) = τ ( x ) = x in (2.10). In 2007, Bae and Park [5] introducedthe functional equation: f ( x + y, z + w ) + f ( x − y, z − w ) = 2 f ( x, z ) + 2 f ( y, w ) . (2.13)The functional equation (2.13) corresponds to σ ( x ) = τ ( x ) = − x in (2.10),where S is an Abelian group. When X = Y = R , the function f : R → R given by f ( x, y ) = ax + bxy + cy is a solution of (2.13) where a, b and c arefixed real numbers. It is clear that when ( z, w ) = ( x, y ) and σ = τ in (2.8),(2.9) and (2.10), we get the functional equations (2.4), (2.5) and (2.6), where g ( x ) := f ( x, x ) for all x ∈ S . In addition, if S is an Abelian group, σ ( x ) = − x and z = w = 0 in (2.8), (2.9) and (2.10), we find the functional equations (2.1),(2.7) and (2.2) with f ( x ) := f ( x,
0) for all x ∈ S. IZ. EL-FASSI General solution of Eq. (2.8)
Theorem 1.
Let ( S, +) be a commutative semigroup, K a field of characteristicdifferent from , and let σ, τ : S → S be involutions. Then, the general solution f : S → K of the d’Alembert type equation: f ( x + y, z + w ) + f ( x + σ ( y ) , z + τ ( w )) = 2 f ( x, z ) f ( y, w ) , x, y, z, w ∈ S, (3.1) is f ( x, y ) = χ ( x, y ) + χ ( σ ( x ) , τ ( y ))2 , x, y ∈ S, (3.2) where χ : S → K is an arbitrary multiplicative function (i.e., χ ( x + y, z + w ) = χ ( x, z ) χ ( y, w ) for x, y, z, w ∈ S ).Proof. Replacing ( y, w ) by ( σ ( y ) , τ ( w )) in (4.10), we get f ( x + y, z + w ) + f ( x + σ ( y ) , z + τ ( w )) = 2 f ( x, z ) f ( σ ( y ) , τ ( w )) (3.3)for all x, y, z, w ∈ S . From (4.10) and (3.3), we obtain f ( σ ( y ) , τ ( w )) = f ( y, w ) , y, w ∈ S, (3.4)and hence f ( σ ( x ) + y, τ ( z ) + w ) = f ( x + σ ( y ) , z + τ ( w )) , x, y, z, w ∈ S. (3.5)We have two cases: f ( x + σ ( y ) , z + τ ( w )) = f ( x + y, z + w ) and f ( x + σ ( y ) , z + τ ( w )) = f ( x + y, z + w ) . Case 1: If f ( x + σ ( y ) , z + τ ( w )) = f ( x + y, z + w ) , then by (4.10), we get f ( x + y, z + w ) = f ( x, z ) f ( y, w ) , x, y, z, w ∈ S, (3.6)i.e., f ( u + v ) = f ( u ) f ( v )with u = ( x, z ) , v = ( y, w ) and u + v = ( x + y, z + w ) . So, f is a multi-plicative function and can be written in the form (4.11): f ( x, y ) = ( f ( x, y ) + f ( σ ( x ) , τ ( y )). Case 2: If f ( x + σ ( y ) , z + τ ( w )) = f ( x + y, z + w ) , then there exist x , y , z , w ∈ S such that f ( x + y , z + w ) − f ( x + σ ( y ) , z + τ ( w )) = 0 . (3.7)In this case, we define the function F : S → K by F ( x, z ) = f ( x + y , z + w ) − f ( x + σ ( y ) , z + τ ( w )) , x, z ∈ S. NEW TYPE OF FUNCTIONAL EQUATIONS 5
By (3.7), we have F ( x , z ) = 0 and from (3.4) and (3.5), we get F ( σ ( x ) , τ ( z )) = − F ( x, z ) and F ( x + σ ( y ) , z + τ ( w )) = − F ( σ ( x ) + y, τ ( z ) + w )(3.8)for all x, y, z, w ∈ S . Also F ( x + y, z + w ) = f ( x + y + y , z + w + w ) − f ( x + y + σ ( y ) , z + w + τ ( w )) F ( x + σ ( y ) , z + τ ( w )) = f ( x + σ ( y ) + y , z + τ ( w ) + w ) − f ( x + σ ( y ) + σ ( y ) , z + τ ( w ) + τ ( w ))for all x, y, z, w ∈ S . Adding these equations and using (3.4), we obtain F ( x + y, z + w )+ F ( x + σ ( y ) , z + τ ( w ))= 2 f ( y, w )[ f ( x + y , z + w ) − f ( x + σ ( y ) , z + τ ( w ))]for all x, y, z, w ∈ S . Hence F ( x + y, z + w ) + F ( x + σ ( y ) , z + τ ( w )) = 2 f ( y, w ) F ( x, z ) , x, y, z, w ∈ S. (3.9)Interchanging x, y and z, w in (3.9), we get F ( x + y, z + w ) + F ( y + σ ( x ) , w + τ ( z )) = 2 f ( x, z ) F ( y, w ) , x, y, z, w ∈ S, and by (3.8), we have F ( x + y, z + w ) − F ( x + σ ( y ) , z + τ ( w )) = 2 f ( x, z ) F ( y, w ) , x, y, z, w ∈ S. (3.10)Adding (3.9) and (3.10), we find F ( x + y, z + w ) = F ( x, z ) f ( y, w ) + F ( y, w ) f ( x, z ) , x, y, z, w ∈ S. (3.11)i.e., F ( u + v ) = F ( u ) f ( v ) + F ( v ) f ( u ) , with u = ( x, z ) , v = ( y, w ) . It follows from this equation that F has the form F ( u ) = χ ( u ) + χ ( u )2 , with u = ( x, z ) , x, z ∈ S, (3.12)where the functions χ , χ : S → K are multiplicative (see [2, 6, 17]).Replacing ( x, z ) by ( x + t, z + s ) in (3.11), we obtain F ( x + t + y, z + s + w ) = F ( x + t, z + s ) f ( y, w ) + F ( y, w ) f ( x + t, z + s ) by(3.11) = [ F ( x, z ) f ( t, s ) + F ( t, s ) f ( x, z )] f ( y, w )+ F ( y, w ) f ( x + t, z + s ) (3.13) IZ. EL-FASSI for all x, y, z, w, s, t ∈ S. Replacing ( y, w ) by ( t + y, s + w ) in (3.11), we get F ( x + t + y, z + s + w ) = F ( x, z ) f ( t + y, s + w ) + F ( t + y, s + w ) f ( x, z ) by(3.11) = F ( x, z ) f ( t + y, s + w )+ [ F ( y, w ) f ( t, s ) + F ( t, s ) f ( y, w )] f ( x, z ) (3.14)for all x, y, z, w, s, t ∈ S. It follows from (3.13) and (3.14) that F ( y, w )[ f ( x + t, z + s ) − f ( x, z ) f ( t, s )] = F ( x, z )[ f ( t + y, s + w ) − f ( t, s ) f ( y, w )]for all x, y, z, w, s, t ∈ S. Setting ( y, w ) = ( x , z ) and h ( t, s ) = F ( x , z ) − [ f ( x + t, z + s ) − f ( x , z ) f ( t, s )].We have f ( x + t, z + s ) − f ( t, s ) f ( x, z ) = F ( x, z ) h ( t, s ) , x, z, t, s ∈ S (3.15)and f ( x + t, z + s ) − f ( t, s ) f ( x, z ) = F ( t, s ) h ( x, z ) , x, z, t, s ∈ S. So, F ( x, z ) h ( t, s ) = F ( t, s ) h ( x, z ) , x, z, t, s ∈ S. Setting x = x and z = z , wehave h ( t, s ) = F ( x , z ) − F ( t, s ) h ( x , z ) . Since K is quadratically closed there exists an α ∈ K such that α = F ( x , z ) − h ( x , z ) . Hence the equation (3.15), with ( y, w ) in place of ( t, s ), becomes f ( x + y, z + w ) = f ( y, w ) f ( x, z ) + α F ( x, z ) F ( y, w ) , x, z, y, w ∈ S. (3.16)Multiplying (3.11) by α and adding to and subtracting it from (3.16), we obtainrespectively f ( x + y, z + w ) + αF ( x + y, z + w ) = [ f ( y, w ) + αF ( y, w )][ f ( x, z ) + αF ( x, z )]and f ( x + y, z + w ) − αF ( x + y, z + w ) = [ f ( y, w ) − αF ( y, w )][ f ( x, z ) − αF ( x, z )] . So the functions χ := f + αF and χ := f − αF are multiplicative, and byaddition we get (3.12).Now in view of (3.4) and (3.8) we have χ ( σ ( x ) , τ ( z )) = f ( σ ( x ) , τ ( z )) + αF ( σ ( x ) , τ ( z )) f ( x, z ) − αF ( x, z ) = χ ( x, z ) , x, z ∈ S. The converse implication follows by simple calculations. This completes theproof of the theorem. (cid:3)
NEW TYPE OF FUNCTIONAL EQUATIONS 7
Remark 2. the commutativity of S can be replaced by the conditions f ( x + t + y, z + s + w ) = f ( x + y + t, z + w + s ) , x, y, z, w, s, t ∈ S (3.17)and f ( x + y, z + w ) = f ( y + x, w + z ) , x, y, z, w ∈ S. (3.18)If S has a neutral element, the second of these conditions is implied by the firstone and can be omitted. Corollary 3 ( [14, Theorem 1]) . Let ( S, +) be a commutative semigroup, K afield of characteristic different from , and let σ : S → S be involution. Then,the general solution g : S → K of the d’Alembert equation: g ( x + y ) + g ( x + σ ( y )) = 2 g ( x ) g ( y ) , x, y ∈ S, is g ( x ) = m ( x ) + m ( σ ( x ))2 , x ∈ S, where m : S → K is an arbitrary multiplicative function (i.e., m ( x + y ) = m ( x ) m ( y ) for x, y ∈ S ).Proof. In Theorem 1, it suffice to take z = x, y = w, τ = σ and g ( x ) := f ( x, x )for all x ∈ S . (cid:3) General solution of Eq. (2.9)
Theorem 4.
Let ( S, +) be a commutative semigroup, G a 2-cancellative abeliangroup, and let σ, τ : S → S be involutions. Then, the general solution f : S → G of the Jensen type equation: f ( x + y, z + w ) + f ( x + σ ( y ) , z + τ ( w )) = 2 f ( x, z ) , x, y, z, w ∈ S, (4.1) is f ( x, y ) = a ( x, y ) + c, x, y ∈ S, (4.2) where c ∈ G is an arbitrary constant and a : S → G is an arbitrary addi-tive function (i.e., a ( x + y, z + w ) = a ( x, z ) + a ( y, w ) for x, y, z, w ∈ S ) with a ( σ ( x ) , τ ( y )) = − a ( x, y ) for all x, y ∈ S .Proof. Replacing ( y, w ) by ( y + σ ( y ) , w + τ ( w )) in (4.1), we obtain f ( x + y + σ ( y ) , z + w + τ ( w )) = f ( x, z ) , x, y, z, w ∈ S. (4.3) IZ. EL-FASSI
Putting ( x, z ) = ( x + u, z + v ) in (4.1), we get f ( x + u + y, z + v + w ) + f ( x + u + σ ( y ) , z + v + τ ( w )) = 2 f ( x + u, z + v )(4.4)for all x, y, z, u, v, w ∈ S . Interchanging u, y and v, w in (4.4), we find f ( x + u + y, z + v + w ) + f ( x + y + σ ( u ) , z + w + τ ( v )) = 2 f ( x + y, z + w )(4.5)for all x, y, z, u, v, w ∈ S . Adding the last two equations and using (4.1), wehave f ( x + u + y, z + v + w ) + f ( x, z ) = f ( x + u, z + v ) + f ( x + y, z + w ) (4.6)for all x, y, z, u, v, w ∈ S . Replacing ( u, v ) by ( σ ( x ) , τ ( z )) in (4.6), we get f ( x + σ ( x ) + y, z + τ ( z ) + w ) + f ( x, z ) = f ( x + σ ( x ) , z + τ ( z )) (4.7)+ f ( x + y, z + w )for all x, y, z, , w ∈ S . From (4.7) and (4.3), we obtain f ( y, w ) + f ( x, z ) = f ( x + σ ( x ) , z + τ ( z )) + f ( x + y, z + w ) (4.8)for all x, y, z, w ∈ S . Interchanging x, y and z, w in (4.8), we have f ( x, z ) + f ( y, w ) = f ( y + σ ( y ) , w + τ ( w )) + f ( x + y, z + w ) (4.9)for all x, y, z, w ∈ S . It follows from (4.8), (4.9) that f ( x + σ ( x ) , z + τ ( z )) = f ( y + σ ( y ) , w + τ ( w )) , x, y, z, w ∈ S. Then F ( x + σ ( x ) , z + τ ( z )) is a constant set, say c ∈ G . So (4.9) gives f ( x, z ) − c + f ( y, w ) − c = f ( x + y, z + w ) − c, x, y, z, w ∈ S. Putting a ( x, z ) := f ( x, z ) − c for x, y ∈ S , so a ( x + y, z + w ) = a ( x, z ) + a ( y, w )for all x, y, z, w ∈ S , this means that a is additive. Putting (4.1) into (4.1)1 wefind a ( σ ( x ) , τ ( y )) = − a ( x, y ) for all x, y ∈ S and this completes the proof. (cid:3) Remark 5.
Here too, the commutativity of S can be replaced by the conditions(3.17) and (3.18).
Corollary 6 ( [14, Theorem 2]) . Let ( S, +) be a commutative semigroup, G be a 2-cancellative abelian group, and let σ : S → S be involution. Then, thegeneral solution g : S → G of the Jensen equation: g ( x + y ) + g ( x + σ ( y )) = 2 g ( x ) , x, y ∈ S, (4.10) NEW TYPE OF FUNCTIONAL EQUATIONS 9 is g ( x ) = ψ ( x ) + a, x ∈ S, (4.11) where a ∈ G is an arbitrary constant and ψ : S → G is an arbitrary additivefunction (i.e., ψ ( x + y ) = ψ ( x ) + ψ ( y ) ) with ψ ( σ ( x )) = − ψ ( x ) for all x ∈ S .Proof. In Theorem 4, it suffice to take z = x, y = w, τ = σ and g ( x ) := f ( x, x )for all x ∈ S . (cid:3) In the proof of the next theorem we shall need the following corollary.
Corollary 7.
Let S , G and σ, τ be as in Theorem 4. Then, the general solution h : S × S → G of the Jensen type equation: h (( x + y, z + w ) , ( t, s )) + h (( x + σ ( y ) ,z + τ ( w )) , ( t, s )) = 2 h (( x, z ) , ( t, s )) ,x, y, z, w, t, s ∈ S, is h (( x, y ) , ( t, s )) = a (( x, y ) , ( t, s )) + c ( t, s ) , x, y, t, s ∈ S, where c : S → G is an arbitrary function and a : S × S → G is an arbitraryfunction additive (i.e., a (( x + y, z + w ) , ( t, s )) = a (( x, z ) , ( t, s )) + a (( y, w ) , ( t, s )) for x, y, z, w, t, s ∈ S ) with a (( σ ( x ) , τ ( y )) , ( t, s )) = − a (( x, y ) , ( t, s )) for all x, y, t, s ∈ S .Proof. Obvious. (cid:3) General solution of Eq. (2.10)
Theorem 8.
Let ( S, +) be a commutative semigroup, H an abelian group,uniquely divisible by 2, and let σ, τ : S → S be involutions. Then, the generalsolution f : S → H of the quadratic type equation: f ( x + y, z + w ) + f ( x + σ ( y ) , z + τ ( w )) = 2 f ( x, z ) + 2 f ( y, w ) , x, y, z, w ∈ S, (5.1) is f ( u ) = B ( u, u ) + T ( u ) , u = ( x, z ) , x, y ∈ S, (5.2) where T : S → H is an arbitrary additive function (i.e., T ( u + v ) = T ( u )+ T ( v ) where u = ( x, z ) , v = ( y, w ) and u + v = ( x + y, z + w ) ) with T ( σ ( x ) , τ ( z )) = T ( x, z ) and B : S × S → H is an arbitrary symmetric biadditive function(i.e., B ( u, v ) = B ( v, u ) and B ( u + v, r ) = B ( u, r ) + B ( v, r ) for u, v, r ∈ S )with B (( σ ( x ) , τ ( z )) , ( y, w )) = − B (( x, z ) , ( y, w )) . Proof.
Replacing ( y, w ) by ( σ ( y ) , τ ( w )) in (5.1), we see that f ( y, w ) = f ( σ ( y ) , τ ( w )) , y, w ∈ S, (5.3)so, f ( x + σ ( y ) , z + τ ( w )) = f ( σ ( x ) + y, τ ( z ) + w ) , x, z, y, w ∈ S. (5.4)Also in (5.1), we replace ( x, z ) first by ( x + t, z + s ) and then by ( x + σ ( t ) , z + τ ( s )), and find f ( x + t + y, z + s + w ) + f ( x + t + σ ( y ) , z + s + τ ( w ))= 2 f ( x + t, z + s ) + 2 f ( y, w ) , x, y, z, w, t, s ∈ S,f ( x + σ ( t ) + y, z + τ ( s ) + w ) + f ( x + σ ( t ) + σ ( y ) , z + τ ( s ) + τ ( w ))= 2 f ( x + σ ( t ) , z + τ ( s )) + 2 f ( y, w ) , x, y, z, w, t, s ∈ S. By subtraction we obtain h (( x + y, z + w ) , ( t, s )) + h (( x + σ ( y ) , z + τ ( w )) , ( t, s )) = 2 h (( x, z ) , ( t, s ))(5.5)for all x, z, t, s, y, w ∈ S, where h (( x, z ) , ( t, s )) : = f ( x + t, z + s ) − f ( x + σ ( t ) , z + τ ( s ))4 (5.6)for all x, z, t, s ∈ S . According to Corollary 7, we have h ( u, r ) = B ( u, r ) + c ( r )where B : S × S → H is an arbitrary function additive, i.e., B ( u + v, r ) = B ( u, r ) + B ( v, r ) for all u, v, r ∈ S (with u = ( x, z ) , v = ( y, w ) , u + v =( x + y, z + w ) and r = ( t, s )) and B (( σ ( x ) , τ ( y )) , r ) = − B (( x, y ) , r ). Hence h (( x, z ) , r ) + h (( σ ( x ) , τ ( z )) , r ) = 2 c ( r ) . On the other hand, from (5.6) and (5.4), we have h ( u, r ) = h ( r, u ) and, inview of (5.3) and (5.4), h (( σ ( x ) , τ ( z )) , r ) = − h (( x, z ) , r ) . So, c ( r ) = 0 and h ( u, r ) = b ( u, r ), that is, b is symmetric and biadditive.Now from (5.1) with ( y, w ) = ( x, z ), we find f (2 x, z ) = 4 f ( x, z ) − f ( x + σ ( x ) , z + τ ( z )) and from (5.5) with ( t, s ) = ( x, z ), we have h (( x, z ) , ( x, z )) = [ f (2 x, z ) − f ( x + σ ( x ) , z + τ ( z ))]. Hence f ( u ) = B ( u, u ) + 12 f ( x + σ ( x ) , z + τ ( z )) . Putting ( x, y ) = ( x + σ ( x ) , y + σ ( y )) and ( z, w ) = ( z + τ ( z ) , w + τ ( w )) in(5.1), we get f ( x + σ ( x ) + y + σ ( y ) , z + τ ( z ) + w + τ ( w )) = f ( x + σ ( x ) , z + τ ( z ))+ f ( y + σ ( y ) , w + τ ( w )) , x, y, z, w ∈ S, NEW TYPE OF FUNCTIONAL EQUATIONS 11 which means that the function T ( x, z ) := f ( x + σ ( x ) , z + τ ( z )) is additive on S . Also T ( σ ( x ) , τ ( z )) = T ( x, z ) for all x, z ∈ S .Conversely, it is easy to check that any function f of the form (5.2) satisfies(5.1). This completes the proof of the theorem. (cid:3) Corollary 9.
Let ( S, +) be a commutative semigroup, H an abelian group,uniquely divisible by 2, and let σ : S → S be involution. Then, the generalsolution g : S → H of the quadratic equation: g ( x + y ) + g ( x + σ ( y )) = 2 f ( x ) + 2 f ( y ) , x, y ∈ S, is f ( x ) = b ( x, x ) + ψ ( x ) , x ∈ S, where ψ : S → H is an arbitrary additive function with ψ ( σ ( x )) = ψ ( x ) and b : S → H is an arbitrary symmetric biadditive function with b ( σ ( x ) , y ) = − b ( x, y ) .Proof. In Theorem 8, it suffice to take z = x, y = w, τ = σ and g ( x ) := f ( x, x )for all x ∈ S . (cid:3) We end the paper with some questions and a conclusion.
Questions 10.
1) As a future work, what are the general solutions of thefollowing equations: f ( x + y + α, z + w + β ) + f ( x + σ ( y ) + α, z + τ ( w ) + β ) = 2 f ( x, z ) f ( y, w ) f ( x + σ ( y ) + α, z + τ ( w ) + β ) − f ( x + y + α, z + w + β ) = 2 f ( x, z ) f ( y, w ) f ( x + y, z + w ) + f ( x + σ ( y ) , z + τ ( w )) = 2 f ( x, z ) g ( y, w ) f ( x + y, z + w ) + g ( x + σ ( y ) , z + τ ( w )) = h ( x, z ) f ( x + y, z + w ) + f ( x + σ ( y ) , z + τ ( w )) = 2 f ( x, z ) + 2 g ( y, w ) f ( x + y, z + w ) + g ( x + σ ( y ) , z + τ ( w )) = 2 f ( x, z ) + 2 h ( y, w )for all α, β, x, y, z, w ∈ S , where S is a semi-group and σ, τ : S → S are involu-tions.2) We can also study the Hyers-Ulam stability of the previous functionalequations. Conclusion
In this work, we managed to find general solutions to the functional equations(2.8), (2.9) and (2.10). From these equations we can derive a new type offunctional equation which will be of utmost importance in the future. Thiscould be a potential future work.
References [1] J. Acz´el,
The general solution of two functional equations by reduction to functionsadditive in two variables and with the aid of Hamel bases , Glas. Mat.-Fiz. Astronom. (1965), 65-73.[2] J. Acz´el and J. Dhombres, Functional Equations in Several Variables , Cambridge Univ.Press, (1989).[3] J. Acz´el, J. K. Chung and C. T. Ng,
Symmetric second differences in product form ongroups , in: Th.M.Rassias(ed.) Topics in Mathematical Analysis, World Scientific Publ.,(1989), 1-22.[4] D. Amir,
Characterization of Inner Product Spaces ,Birkh¨auser Verlag, (1986).[5] J.-H. Bae, W.-P. Park,
A functional equation originating from quadratic forms , J.Math.Anal. Appl. (2007), 1142-1148.[6] J. K. Chung, Pl. Kannappan and C. T. Ng,
A generalization of the cosine-sine functionalequation on groups , Linear Algebra Appl. (1985), 259-277.[7] S. Czerwik, Functional Equations and Inequalities in Several Variables , World Scientific,Singapore (2002).[8] J. d’Alembert,
Addition au Memoire sur la courbe que forme une corde tendue mise envibration , Histoire de l’Academie Royale, pp. 355-360, (1750).[9] T.M.K. Davison,
D’Alembert’s functional equation on topological groups , Aequat. Math. (2008), 33-53.[10] T.M.K. Davison, D’Alembert’s functional equation on topological monoids . Publ. Math.Debrecen. (2009), 41-66.[11] P. Jordan and J. von Neumann, On inner products in linear metric spaces , Ann. Math. (1935), 719-723.[12] P. Kannappan, Functional Equations and Inequalities with Applications , Springer, Berlin(2009).[13] P.K. Sahoo, P. Kannappan,
Introduction to Functional Equations , CRC Press, BocaRaton (2011).[14] P. Sinopoulos,
Functional equations on semigroups , Aequat. Math. (2000) 255-261.[15] H. Stetkær, Functional equations on abelian groups with involution , Aequat.Math. (1-2), (1997), 144-172.[16] H. Stetkær, Functional Equations on Groups , World Scientific, Singapore, (2013).[17] E. Vincze,
Eine allgemeinere Methode in der Theorie der Funktionalgleichungen, II ,Publ. Math. Debrecen (1962), 314-323. Department of Mathematics,Faculty of Sciences and Techniques,Sidi Mohamed Ben Abdellah University, B.P. 2202Fez, Morocco
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