SciELO - Scientific Electronic Library Online

 
vol.36 número3Hyperstability of cubic functional equation in ultrametric spacesSix dimensional matrix summability of triple sequences índice de autoresíndice de materiabúsqueda de artículos
Home Pagelista alfabética de revistas  

Servicios Personalizados

Revista

Articulo

Indicadores

Links relacionados

  • En proceso de indezaciónCitado por Google
  • No hay articulos similaresSimilares en SciELO
  • En proceso de indezaciónSimilares en Google

Compartir


Proyecciones (Antofagasta)

versión impresa ISSN 0716-0917

Proyecciones (Antofagasta) vol.36 no.3 Antofagasta set. 2017

http://dx.doi.org/10.4067/S0716-09172017000300485 

Articles

A generalization of variant of Wilson's type Hilbert space valued functional equations

Hajira Dimou1 

Samir Kabbaj2 

1 Ibn Tofail University, Faculty of Sciences, Department of Mathematics, Kenitra, Morocco, e-mail: dimouhajira@gmail.com

2 Ibn Tofail University, Faculty of Sciences, Department of Mathematics, Kenitra, Morocco, e-mail: samkabbaj@yahoo.fr

Abstract:

In the present paper we characterize, in terms of characters, multiplicative functions, the continuous solutions of some functional equations for mappings defined on a monoid and taking their values in a complex Hilbert space with the Hadamard product. In addition, we investigate a superstability result for these equations.

Keywords: D'Alembert's functional equation; Hilbert space; Hadamard product; superstability.

1.Introduction

Let M be a monoid i.e., is a semigroup with an identify element that we denote by e and σ,τ: M → M are two involutive automorphisms. That is σ(xy) = σ(x)σ(y), τ(xy) = τ(x)τ(y) and σ(σ(x)) = x, τ(τ(x)) = x for all x,yM. By a variant of Wilson's functional equation on M we mean the functional equation

(1.1) f((y)) + f(τ(y)x) = 2f(x)g(y), x,yM,

Where f, g:MC are the unknown functions. A special case of Wilson's functional equation is d'Alembert's functional equation:

(1.2) f((y)) + f(τ(y)x) = 2f(x)f(y), x,yM

The solutions of equation (1.2) are known(2). Further contextual and historical discussion on the functional equation (1.1) and (1.2) can be found, e.g., in (6.2).

The present paper studies an extension to a situation where the unknown functions f, g map a possibly non-abelian group or monoid into a complex Hilbert space H with the Hadamard product. Our considerations refer mainly to results by Rezaei (4), Zeglami (11).

It has been proved (3) that the functional equation (1.2) with σ = id is superstable in

the class of functions f:GC, if every such function satisfies the inequality

|f(xy) + f(τ(y)x) - 2f(x)f(y)| ≤ ϵ f or all x,yG,

where ϵ is a fixed positive real number. Then either f is a bounded function or

f(xy) + f(τ(y)x) = 2f(x)f(y), x,yG.

Let H be a separable Hilbert space with a orthonormal basis { en , nN}. For two vectors x,yH, the Hadamard product, also known as the entrywise product on the Hilbert space H is defined by

The Cauchy-Schwarz inequality together with the Parseval identity ensure that the Hadamard multiplication is well defined. In fact,

The purpose of this work is first to give a characterization, in terms of multiplicative functions, the solutions of the Hilbert space valued functional equation by Hadamard product:

(1.5) f((y)) + f(τ(y)x) = 2g(x) * f(y), x,yM.

When f we determine the solutions of the functional equation

(1.6) f((y)) + f(τ(y)x) = 2f(x) * g(y), x,yM,

where f,g:MH are the unknown functions. Second, we determine a characterization of the following d'Alembert-Hilbert-valued functional equation:

(1.7) f((y)) + f(τ(y)x) = 2f(x) * f(y), x,yM.

Throughout the paper, N, R and C stand for the sets of positive integers, real numbers and complex numbers, respectively. We let G denote a group and S denote a semigroup i.e., a set with an associative composition rule.

A function A:MC is called additive, if it satisfies A(xy) = A(x) + A(y) for all x,yM.

A multiplicative function on M is a map χ:MC such that χ(xy) = χ(x) χ(y) for all x,yM.

A monoid M is generated by its squares if for every for some x1,x2,…,xn M.

A character on a group G is a homomorphism from G into the multiplicative of non-zero complex numbers. While a non-zero multiplicative function on a group can never take the value 0, it is posible for a multiplicative function on a monoid M to take the value 0 on a proper, non-empty subset of M. If χ:M → C is multiplicative and χ ≠ 0, then

Iχ = {xM / χ(x) = 0}

is either empty or a proper subset of M. The fact that χ is multiplicative establishes that

is a two-sided ideal in M if not empty (for us an ideal is never the empty set). It follows also that M\ is a subsemigroup of M.

Let C(M) denote the algebra of continuous functions from M into C.

2.Solutions of (1.5) and (1.6)

In this section, we solve the functional equation (1.5) by expressing its solutions in terms of

multiplicative functions.

Theorem 2.1. Let M be a monoid, let σ, τ:MM be involutive automorphisms. Assume

that the functions f, g:MH satisfy (1.5). Then, there exists a positive integer N such that

for all xM and k > 0. Furthermore, for every k ∈ {1,2,….,N}, we have the following possibilities:

for all xM, where χk is a non-zero multiplicative function of M such that χk ° σ ° τ = χk ° τ ° σ and α k C \ {0}. If M is a topological monoid and fC(M), then χk, χk ° σ ° τ C (M)

Proof. For every integer k ≥ 0, consider the functions f k,g k:MC defined by

If we put y = e in (2.1), we find that fk (x) = fk (e) gk (x). So, if we take αk = f k(e),

equation (2.1) can be written as follows:

αkgk ((y)) + αkgk (τ(y)x) = 2 αkgk (y) f or all x,yM.

Then, either αk = 0 or g k is a solution of equation (1.6). In view of ((2)), Theorem 3.2), one of the following statements holds:

(a) We have that

f k = 0 and g k is an arbitrary function.

(b) There exists a multiplicative function χ k such that

If H is infinite-dimensional, then

for every xM. Since g k(e) = 1, statement (b) is not possible for infinitely many positive integers k. Hence, there exists some positive integer N such that f k = 0 for every k > N. Thus, g k is an arbitrary function for any k > N, f can be represented as

and the expressions of the component functions f n and g n, 1 ≤ nN, of f and g come from statements (a) and (b) above. In the case where H is finite- dimensional, the proof is clear.

As a consequence of Theorem 2.1 we derive formulas for the solutions of d'Alembert's Hilbert space valued functional equation (1.7).

Corollary 2.2. Let M be a monoid, let σ,τ:MM be involutive automorphisms. Assume

that the functions g:MH satisfy (1.7). Then, there exists a positive integer N such that

for all xM and k > 0. Furthermore, for every k ∈ {1,2,….,N}, such that

where ϵk = 1 or 0 for every k ∈ {1,2,…..,N}. for all x ∈ M, where χ k is a non-zero multiplicative function of M such that χ k ° σ ° τ = χk ° τ ° σ

If M is a topological monoid and fC(M), then χk, χk ° σ ° τC(M).

Proof. The proof follows by putting f = g in Theorem 2.1.

Corollary 2.3. Let M be a monoid, let τ: MM be involutive automorphisms. Assume that

the functions f, g; MH satisfy

f(xt) + f(τ(y)x) = 2g(x) * f(y).

Then, there exists a positive integer N such that

for all x ∈ M and k > 0. Furthermore, for every k ∈ {1,2,….,N}, we have the following possibilities:

for all x ∈ M, where χk is a non-zero multiplicative function of M and αk C \ {0}.

If M is a topological monoid and fC(M), then χk, χk ° τ C(M).

Proof. The proof follows by putting σ = id in Theorem 2.1.

We complete this section with a result concerning Wilson Hilbert space valued functional equation (1.6).

Theorem 2.4. Let M be a monoid which is generated by its squares, let σ,τ: MM be involutive automorphisms. Assume that the pair f,g: MC, satisfy Wilson's Hilbert valued functional equation (1.6). Then, there exists a positive integer N such that

for all x ∈ M and k > 0. Furthermore, for every k ∈ {1,2,….,N}, we have the following possibilities:

Where χk: M C is a non-zero multiplicative function with χk ° σ ° τ = χk ° τ ° σ and for some αkC \ {0}.

(ii) There exists a non-zero multiplicative function χk: M C with with χk ° σ ° τ = χk ° τ ° σ such that

Furthermore, we have

(1) If χk ≠ χk ° σ ° τ, then

for some αk , βkC \ {0}.

(2) If χk = χk ° σ ° τ, then there exists a non-zero additive function Ak: M \ Iχk°σ → C with A k ° τ = -A k ° σ such that

for some αk , ∈ C.

Conversely, if f and g have the forms described above, then the pair (f,g) is a solution of equation (1.6). Moreover, if M is a topological monoid generated by its squares, and f, gC (M), then χk, χk ° σ, χk ° τ, χk ° σ ° τ C(M), while A kC(M\I χk°σ ).

Proof. We proceed as in the proof of Theorem 2.1. For every integer k ≥ 0, we consider the functions f k,g k:MC, defined by

Since the pair (f,g) satisfies (1.6), for all kN we have

By (6),Theorem 3.4) we infer that there are only the following cases

for some αk , ∈ C. Conversely, the functions given with properties satisfy the functional equation (2.2). The continuation of the proof depends on the dimension of H. In fact, if H is infinite-dimensional, then

⟨(g)/x),ex⟩ = gk(x) → 0 as k → + ∞

for every xM. Statements (b) and (c) are not possible for infinitely positive integers n. Hence, there exists some positive integer N such that f k = 0 for every k > N. Thus, f can be represented as

g k is an arbitrary function for any k > N, and expressions of the component functions f n and gn , 1 ≤ nN of f and g follow from the previous discussion. In the case where H is a finite-dimensional space, the proof is clear.

Corollary 2.5. Let M be a monoid which is generated by its squares, let τ:M → be an involutive automorphism, and let the pair f,g: MH satisfy the functional equation

f(xy) + f(τ(y)x) = 2f(x) * g(y), x, yM.

Then, there exists a positive integer N such that

for all xM and k > 0. Furthermore, for every k ∈ {1,2,….,N} we have the following possibilites:

(2) If χk = χk ° τ, then there exists an additive function Ak:M\I χk C with Ak ° τ = -Ak such that

for some αkC.

Conversely, if f and g have the forms described above, then the pair (f,g) is a solution. Moreover, if M is a topological monoid generated by its squares, and f,gC(M), then χk, χk ° τ ∈ C(M), while A kC( M\Iχk ).

Proof. The proof follows by putting σ = id in Theorem 2.4.

3.Superstability of Hilbert valued cosine type functional equations

The main result of this section is Theorem 3.3 that contains a superstability result for the functional equation (1.6). For the proof of our result we will begin by pointing out a superstability result for the equation

(3.1) f(xy) + f(σ(y)x) = 2f(x)g(y)

where f, g:GC are the unknown functions.

Proposition 3.1. Let δ > 0 be given, let M be a monoid and let σ is an involutive morphism of M. Assume that the functions f,g:MC satisfies the inequality

and that g is unbounded. Then, the ordered pair (f, g) satisfies equation (3.1).

Proof. The proof is part of the proof of (3 ,Theorem 2.1 and Theorem 3.7) if we put χ = 1 that deals with M being a group.

Corollary 3.2. Let δ > 0 be given and let G be a monoid. Assume that the function f:GC satisfies the inequality

where μ is a multiplicative function.

Proof. The proof follows immediately from Propositon 3.1 and Theorem (1, Theorem 4).

Theorem 3.3. Let δ > 0 be given and let M be a monoid. Assume that the functions f, g: MH satisfy the inequality

(3.2) ||f(xy) + f(σ(y)x) - 2f(x)*g(y)||≤ δ f or all x,yM.

Then, either

(i) there exists k ≥ 1 such that the function x ↦ ⟨g(x), ek ⟩ is bounded, or

(ii) the pair (f, g) is a solution of the functional equation:

(3.3.) f(xy) + f(σ(y)x) = 2f(x) * g(y).

Proof. Suppose that the pair (f, g) satisfies (3.2). By applying the Parseval identity and the definition of Hadamard product with the inequality (3.2), we find that the scalar valued functions f k, g k defined by

f k(x) = ⟨f(x), ek ⟩ and gk (x) = ⟨g(x), ek ⟩ f or xM,

satisfy the inequality

|f k(xy) + f k(σ(y)x) - 2f k(x)g k(y)| ≤ δ f or all x,yM.

According to Proposition 3.1, for all kN, we have that either the function x ↦ ⟨g(x), ek ⟩ is bounded or the pair ( fk , gk ) is a solution of (3.1). Then, we conclude that the pair (f, g) satisfies equation (3.3) if assertion (i) fails.

In ((4)) it was proved that if g: HH is surjective, then every component function x ↦ ⟨g(x), en

is unbounded. By applying Theorem (3.3), this leads to the following result.

Corollary 3.4. Let δ > 0 be given. Assume that functions f, g: HH, where g is surjective, satisfy the inequality

||f(xy) + f(σ(y)x) - 2f(x)*g(y)|| ≤ δ for all x,yH.

Then, the pair (f, g) satisfies the equation

f(xy) + f(σ(y)x) = 2f(x)*g(y) for all x,yH.

Proof. Since g is surjective, then every component function x ↦ ⟨g(x), en ⟩ is unbounded. Thus, the proof follows immediately from Theorem 3.3.

Corollary 3.5. Let δ > 0 be given and let G be a topological group. Assume that the function g:GH satisfies the inequality

||g(xy) + g(σ(y)x) - 2g(x) * g(y)|| ≤ δ f or all x,y ∈ G.

Then, either there exists k ≥ 1 such that

or there exist a multiplicative function χ k: MC \ {0} and a positive integer N such that

where ϵn = 1 or 0 for every n ∈ {1,2,….., N}.

Proof. If we put f = g in Theorem 3.3, we immediately have that either there exists k ≥ 1 such that the function x ↦ ⟨g(x),e k⟩ is bounded or g is a solution of the equation

g(xy) + g(σ(y)x) = 2g(x) * g(y), x,yG.

The remainder of the proof follows if we put χ = 1 from Corollary (3), Corollary 3.8) and Corollary 2.3.

Corollary. 3.6. Let δ > 0 be given and let G be a group with identity element. Let g:GH such that

||g(xy) + 2g(x) * g(y)|| ≤ δ f or all x,y ∈ G. Then either g is bounded or g is multiplicative.

Proof. From Corollary 2.2 and Corollary 2.5 and then using (3, Corollary 3.9).

References

[1] J. A. Baker, The stability of the Cosine Equation, Proc. Amer. Math. Soc. 80, 3, pp. 411-416, (1980). [ Links ]

[2] A. Chahbi, B. Fadli, S. Kabbaj, A generalization of the symmetrized multiplicative Cauchy equation. Acta Math. Hung. 149 (1), pp. 170-176, (2016). [ Links ]

[3] E. Elqorachi and A. Redouani, Solutions and stability of variant of Wilson's functional equation (14 May 2015), arXiv:1505.06512v1 (math. CA). Demonstratio Math. (to appear). [ Links ]

[4] H. Rezaei and M. Sharifzadeh, On the super-stability of exponential hilbert valued functional equations, J. Inequal. Appl. Article ID 114. (2011). [ Links ]

[5] L. Székelyhidi, D'Alembert's functional equation on compact groups, Banach J. Math. Anal. 1, no. 2, pp. 221-226, (2007). [ Links ]

[6] KH. Sabour, B. Fadli and S. Kabbaj, Wilson's Functional equation on monoids with involutive automorphisms. Aequationes. Math. 90, no. 5, pp. 189-196, (2016). [ Links ]

[7] H. Stetkaer, D'Alembert's and Wilson's functional equations on step 2 nilpotent groups, Aequationes. Math. 67, No. 3, pp. 1001-1011, (2004). [ Links ]

[8] H. Stetkaer, Functional Equations on Groups, World Scientific Publishing Co., Singapore, (2013). [ Links ]

[9] H. Stetkaer, A link between Wilson's functional and d'Alembert's functional equations, Aequationes. Math. 90, No. 2, pp. 407-409, (2016). [ Links ]

[10] W. H. Wilson, On a certain related functional equations, Proc. Amer. Math. Soc. 26, pp. 300-312, (1920). [ Links ]

[11] D. Zeglami, M. Tial and B. Fadli, Wilson's type Hilbert space valued functional equations. Adv. Pure Appl. Math. 7 (3), pp. 189-196, (2016). [ Links ]

Received: December 2016; Accepted: May 2017

Creative Commons License This is an open-access article distributed under the terms of the Creative Commons Attribution License