A Variant of D’Alembert’s Functional Equation on Semigroups with Endomorphisms
Let S be a semigroup, and let φ, ψ: S → S be two endomorphisms (which are not necessarily involutive). Our main goal in this paper is to solve the following generalized variant of d’Alembert’s functional equation f(xϕ(y))+f(ψ(y)x)=2f(x)f(y), x,y ∈ S,f\left( {x\varphi \left( y \right)} \right) +...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Sciendo
2022-03-01
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| Series: | Annales Mathematicae Silesianae |
| Subjects: | |
| Online Access: | https://doi.org/10.2478/amsil-2022-0004 |
| _version_ | 1818139860486062080 |
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| author | Akkaoui Ahmed El Fatini Mohamed Fadli Brahim |
| author_facet | Akkaoui Ahmed El Fatini Mohamed Fadli Brahim |
| author_sort | Akkaoui Ahmed |
| collection | DOAJ |
| description | Let S be a semigroup, and let φ, ψ: S → S be two endomorphisms (which are not necessarily involutive). Our main goal in this paper is to solve the following generalized variant of d’Alembert’s functional equation
f(xϕ(y))+f(ψ(y)x)=2f(x)f(y), x,y ∈ S,f\left( {x\varphi \left( y \right)} \right) + f\left( {\psi \left( y \right)x} \right) = 2f\left( x \right)f\left( y \right),\,\,\,\,\,\,x,y\, \in \,S,
where f : S → ℂ is the unknown function by expressing its solutions in terms of multiplicative functions. Some consequences of this result are presented. |
| first_indexed | 2024-12-11T10:34:48Z |
| format | Article |
| id | doaj.art-96738902d9d845bcacb5b9553bae9bc3 |
| institution | Directory Open Access Journal |
| issn | 2391-4238 |
| language | English |
| last_indexed | 2024-12-11T10:34:48Z |
| publishDate | 2022-03-01 |
| publisher | Sciendo |
| record_format | Article |
| series | Annales Mathematicae Silesianae |
| spelling | doaj.art-96738902d9d845bcacb5b9553bae9bc32022-12-22T01:10:45ZengSciendoAnnales Mathematicae Silesianae2391-42382022-03-0136111410.2478/amsil-2022-0004A Variant of D’Alembert’s Functional Equation on Semigroups with EndomorphismsAkkaoui Ahmed0El Fatini Mohamed1Fadli Brahim2Department of Mathematics, Faculty of Sciences, Ibn Tofail University, BP: 14000, Kenitra, MoroccoDepartment of Mathematics, Faculty of Sciences, Ibn Tofail University, BP: 14000, Kenitra, MoroccoDepartment of Mathematics, Faculty of Sciences, Chouaib Doukkali University, BP: 24000, El Jadida, MoroccoLet S be a semigroup, and let φ, ψ: S → S be two endomorphisms (which are not necessarily involutive). Our main goal in this paper is to solve the following generalized variant of d’Alembert’s functional equation f(xϕ(y))+f(ψ(y)x)=2f(x)f(y), x,y ∈ S,f\left( {x\varphi \left( y \right)} \right) + f\left( {\psi \left( y \right)x} \right) = 2f\left( x \right)f\left( y \right),\,\,\,\,\,\,x,y\, \in \,S, where f : S → ℂ is the unknown function by expressing its solutions in terms of multiplicative functions. Some consequences of this result are presented.https://doi.org/10.2478/amsil-2022-0004functional equationd’alembertsemigroupmultiplicative functionendomorphism39b52 |
| spellingShingle | Akkaoui Ahmed El Fatini Mohamed Fadli Brahim A Variant of D’Alembert’s Functional Equation on Semigroups with Endomorphisms Annales Mathematicae Silesianae functional equation d’alembert semigroup multiplicative function endomorphism 39b52 |
| title | A Variant of D’Alembert’s Functional Equation on Semigroups with Endomorphisms |
| title_full | A Variant of D’Alembert’s Functional Equation on Semigroups with Endomorphisms |
| title_fullStr | A Variant of D’Alembert’s Functional Equation on Semigroups with Endomorphisms |
| title_full_unstemmed | A Variant of D’Alembert’s Functional Equation on Semigroups with Endomorphisms |
| title_short | A Variant of D’Alembert’s Functional Equation on Semigroups with Endomorphisms |
| title_sort | variant of d alembert s functional equation on semigroups with endomorphisms |
| topic | functional equation d’alembert semigroup multiplicative function endomorphism 39b52 |
| url | https://doi.org/10.2478/amsil-2022-0004 |
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