Article ID: MTJPAM-D-22-00022

Title: Trigonometric functional equations on non-abelian semigroups

Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-22-00022; Volume 4 / Issue 3 / Year 2022, Pages 175-193

Document Type: Research Paper

Author(s): Belfakih Keltouma a , Elqorachi Elhoucien b

aMultidisciplinary faculty, Ibn Zohr University, Taroudant, Morocco

bDepartment of Mathematics, Ibn Zohr University, Faculty of Sciences, Agadir, Morocco

Received: 27 June 2022, Accepted: 30 September 2022, Published: 29 November 2022.

Corresponding Author: Elqorachi Elhoucien (Email address:

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Let S be a semigroup, and let μ : S → ℂ be a multiplicative function such that μ(xσ(x)) = 1 for all x ∈ S. We study the properties of the solutions of the functional equations

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

where σ is an involutive morphism. The solutions are expressed by means of solutions of d’Alembert’s μ-functional equation and the functional equation

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

in which ϕ is a solution of d’Alembert’s μ-functional equation. As an application we prove that, in a nilpotent group G which is generated by its squares, the solutions of the functional equation

f(xy)+μ(y)f(xy−1)=2f(x)g(y)+2f(y)g(x), x, y ∈ G

are abelian.
We also find the solutions of the functional equation

f(xy)+μ(y)f(xσ(y)) = 2f(x)ϕ(y)+2f(y)ψ(x), x, y ∈ S,

where σ is an involutive anti-automorphism, f : S → ℂ is the unknown function and ϕ, ψ are non-zero solutions of d’Alembert’s μ-functional equation. This enables us to solve the pexider functional equation

f(xy)+μ(y)f(xσ(y)) = 2g1(x)h1(y)+2ψ(x)h2(y), x, y ∈ S

in which f, g1, h1, h2 : S → ℂ are the unknown functions and g1 is even.

Keywords: Semigroup, Involutive automorphism, Involutive anti-automorphism, Multiplicative function, D’Alembert equation, Additive function, Quadratic equation, Pexider equation

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