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