A System of Cosine-Sine Functional Equations on a Semigroup Generated by Its Squares

Ajebbar Omar and Elqorachi Elhoucien
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Ajebbar Omar: Sultan Moulay Slimane University, Multidisciplinary Faculty, Department of Mathematics
Elqorachi Elhoucien: Ibn Zohr University, Faculty of Sciences, Department of Mathematics

A chapter in Functional Equations and Ulam’s Problem, 2026, pp 1-40 from Springer

Abstract: Abstract Given a semigroup S generated by its squares, we determine the complex-valued solutions of the following system of cosine-sine functional equations f ( xy ) = f ( x ) g 1 ( y ) + g 1 ( x ) f ( y ) + λ 1 2 h ( x ) h ( y ) , x , y ∈ S , h ( xy ) = h ( x ) g 2 ( y ) + g 2 ( x ) h ( y ) + λ 2 2 f ( x ) f ( y ) , x , y ∈ S , $$\displaystyle \begin{aligned} \begin {array}{l}f(xy)=f(x)g_{1}(y)+g_{1}(x)f(y)+\lambda _{1}^{2}\,h(x)h(y),\; x,y\in S,\\ h(xy)=h(x)g_{2}(y)+g_{2}(x)h(y)+\lambda _{2}^{2}\,f(x)f(y),\; x,y\in S, \end {array} \end{aligned} $$ where λ 1 , λ 2 ∈ ℂ $$\lambda _{1},\lambda _{2}\in \mathbb {C}$$ are given constants and f , g 1 , g 2 , h : S → ℂ $$f,g_{1},g_{2},h:S\to \mathbb {C}$$ are unknown functions.

Keywords: Semigroup; Cosine-sine equation; Multiplicative function; Additive function; Primary 39B52; Secondary 39B32 (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-032-08949-6_1

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DOI: 10.1007/978-3-032-08949-6_1

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