A generalization of the symmetrized multiplicative Cauchy equation

Chahbi, Abdellatif; Fadli, Brahim; Kabbaj, Samir

doi:10.1007/s10474-016-0584-3

Cited by 11 publications

(6 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the present work we will solve (1) on a general monoid. Other pertinent literature includes Chahbi et al [5], who treated the special case f = h = k of (1) with σ • τ = τ • σ on semigroups.…”

Section: (): V-volmentioning

confidence: 99%

Around the Sine Addition Law and d’Alembert’s Equation on Semigroups

Ebanks

2021

Results Math

View full text Add to dashboard Cite

We treat several related trigonometric functional equations on semigroups. First we make a small improvement to the description of solutions of the sine addition law f (xy) = f (x)g(y) + g(x)f (y) for f, g : S → C, where S is a semigroup. Next we solve the partial Pexiderizationwhere σ, τ are involutive automorphisms of M . The solutions of the latter two equations were given by Ng, Zhao, and Lin assuming that M is a group, then by the author assuming that the monoid M is either regular or generated by its squares. Here we find that new solutions appear on monoids that are neither regular nor generated by their squares. In addition we give the continuous solutions of all three functional equations on topological semigroups/monoids.

show abstract

Section: (): V-volmentioning

confidence: 99%

Around the Sine Addition Law and d’Alembert’s Equation on Semigroups

Ebanks

2021

Results Math

View full text Add to dashboard Cite

show abstract

“…which was solved in [4] by Chahbi et al (σ, τ are two involutive automorphisms). This last equation generalizes Eq (2.2).…”

Section: Introductionmentioning

confidence: 99%

A Variant of D’Alembert’s Functional Equation on Semigroups with Endomorphisms

Akkaoui

Fatini

Fadli

2022

Annales Mathematicae Silesianae

View full text Add to dashboard Cite

Let S be a semigroup, and let φ, ψ: S → S be two endomorphisms (which are not necessarily involutive). Our main goal in this paper is to solve the following generalized variant of d’Alembert’s functional equation f ( x ϕ ( y ) ) + f ( ψ ( y ) x ) = 2 f ( x ) f ( y ) , x , y ∈ S , f\left( {x\varphi \left( y \right)} \right) + f\left( {\psi \left( y \right)x} \right) = 2f\left( x \right)f\left( y \right),\,\,\,\,\,\,x,y\, \in \,S, where f : S → ℂ is the unknown function by expressing its solutions in terms of multiplicative functions. Some consequences of this result are presented.

show abstract

“…x, y ∈ S, and the variant (2) is that τ is an antihomomorphism (on a group typically the group inversion). Some information, applications and numerous references concerning (1), (2) and their further generalizations can be found e.g. in [1,5,8,9].…”

Section: Introductionmentioning

confidence: 99%

The solution of a class functional equations on semi-groups

EL-Fassi

2017

Filomat

View full text Add to dashboard Cite

Let S be a semi-group, and let σ, τ ∈ Antihom(S, S) satisfy τ • τ = σ • σ = id. We show that any solutions f : S → C of the functional equation f (xσ(y)) + f (τ(y)x) = 2 f (x) f (y), x, y ∈ S, has the form f = (m + m • σ • τ)/2, where m is a multiplicative function on S. 1. Set Up and Notation Throughout the paper we work in the following framework: S is a semi-group (a set equipped with an associative composition rule (x, y) → xy) and σ, τ : S → S are two antihomomorphisms (briefly σ, τ ∈ Antihom(S, S)) satisfying τ • τ = σ • σ = id. For any function f : S → C we say that f is σ-even (resp. τ-even) if f • σ = f (resp. f • τ = f), also we use the notationf (x) = f (x −1) in the case S is a group. We say that a function m : S → C is multiplicative, if m(xy) = m(x)m(y) for all x, y ∈ S. If S is a topological space, then we let C(S) denote the algebra of continuous functions from S into C.

show abstract

A generalization of the symmetrized multiplicative Cauchy equation

Cited by 11 publications

References 8 publications

Around the Sine Addition Law and d’Alembert’s Equation on Semigroups

Around the Sine Addition Law and d’Alembert’s Equation on Semigroups

A Variant of D’Alembert’s Functional Equation on Semigroups with Endomorphisms

The solution of a class functional equations on semi-groups

Contact Info

Product

Resources

About