**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}

^{a}Multidisciplinary faculty, Ibn Zohr University, Taroudant, Morocco

^{b}Department 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: elqorachi@hotmail.com)

**Full Text:** PDF

**Abstract**

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

*f*(*x**y*)+*μ*(*y*)*f*(*x**σ*(*y*)) = 2*f*(*x*)*g*(*y*)+2*f*(*y*)*g*(*x*), *x*, *y* ∈ *S*,

*f*(*x**y*)+*μ*(*y*)*f*(*x**σ*(*y*)) = 2*f*(*x*)*f*(*y*)+2*g*(*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*(*x**y*)+*μ*(*y*)*f*(*x**σ*(*y*)) = 2*f*(*x*)*ϕ*(*y*)+2*f*(*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*(*x**y*)+*μ*(*y*)*f*(*x**y*^{−1})=2*f*(*x*)*g*(*y*)+2*f*(*y*)*g*(*x*), *x*, *y* ∈ *G*

are abelian.

We also find the solutions of the functional equation

*f*(*x**y*)+*μ*(*y*)*f*(*x**σ*(*y*)) = 2*f*(*x*)*ϕ*(*y*)+2*f*(*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*(*x**y*)+*μ*(*y*)*f*(*x**σ*(*y*)) = 2*g*_{1}(*x*)*h*_{1}(*y*)+2*ψ*(*x*)*h*_{2}(*y*), *x*, *y* ∈ *S*

in which *f*, *g*_{1}, *h*_{1}, *h*_{2} : *S* → ℂ are the unknown functions and *g*_{1} is even.

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

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