D’Alembert’s Functional Equations on Monoids with an Anti-endomorphism

Let M be a topological monoid. Our main goal is to introduce the functional equation g ( x y ) + μ ( y ) g ( x ψ ( y ) ) = 2 g ( x ) g ( y ) , x , y ∈ M , where ψ : M → M is a continuous anti-endomorphism that need not be involutive and μ : M → C is a continuous multiplicative function such that μ (...

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Bibliographic Details
Published in:Resultate der Mathematik 2020-04, Vol.75 (2), Article 74
Main Authors: Ayoubi, Mohamed, Zeglami, Driss
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
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Summary:Let M be a topological monoid. Our main goal is to introduce the functional equation g ( x y ) + μ ( y ) g ( x ψ ( y ) ) = 2 g ( x ) g ( y ) , x , y ∈ M , where ψ : M → M is a continuous anti-endomorphism that need not be involutive and μ : M → C is a continuous multiplicative function such that μ ( x ψ ( x ) ) = 1 for all x ∈ M . We exploit results on the pre-d’Alembert functional equation by Davison (Publ Math Debrecen 75(1–2):41–66, 2009) and Stetkær (Functional equations on groups. World Scientific Publishing Company, Singapore (2013)) to prove that the continuous solutions g : M → C of this equation can be expressed in terms of multiplicative functions and characters of 2-dimensional representations of M . Interesting consequences of this result are presented.
ISSN:1422-6383
1420-9012
DOI:10.1007/s00025-020-01199-z