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: elqorachi@hotmail.com)

Full Text: PDF

Abstract

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

References:
  1. O. Ajebbar and E. Elqorachi, The cosine-sine functional equation on a semigroup with an involutive automorphism, Aequationes Math. 91, 1115–1146, 2017.
  2. J. K. Chung, Pl. Kannappan and C. T. Ng, A generalization of the cosine-sine functional equation on groups, Linear Algebra Appl. 66, 259–277, 1985.
  3. J. K. Chung, Pl. Kannappan and C. T. Ng, On two trigonometric functional equations, Mathematics Reports Toyama University 11, 153–165, 1988.
  4. B. Ebanks and H. Stetkær, D’Alembert’s other functional equation on monoids with an involution, Aequationes Math. 89, 187–206, 2015.
  5. P. de Place Friis and H. Stetkær, On the cosine-sine functional equation on groups, Aequationes Math. 64 (1-2), 145–164, 2002.
  6. H. Stetkær, Functional equations on abelian groups with involution, Aequations Math. 54, 144–172, 1997.
  7. H. Stetkær, Trigonometric functional equations of rectangular type, Aequationes Math. 56 (3), 251–270, 1998.
  8. H. Stetkær, Functional equations on groups, World Scientific Publishing Co., Singapore, 2013.
  9. H. Stetkær, Extentions of the sine addition law on groups, Aequationes Math. 93 (2), 467–484, 2019.
  10. H. Stetkær, More about Wilson’s functional equation, Aequationes Math. 94, 429–446, 2020.