<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>Let <italic>S</italic> be a semigroup, and let <italic>φ, ψ</italic>: <italic>S</italic> → <italic>S</italic> 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
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2022-0004_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mi>x</mi><mi>ϕ</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mrow><mi>ψ</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mi>x</mi></mrow><mo>)</mo></mrow><mo>=</mo><mn>2</mn><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>f</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>,</mo><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi>x</mi><mo>,</mo><mi>y</mi><mi> </mi><mo>∈</mo><mi> </mi><mi>S</mi><mo>,</mo></mrow></math>
<tex-math>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,</tex-math>
</alternatives>
</disp-formula>
where <italic>f</italic> : <italic>S</italic> → ℂ is the unknown function by expressing its solutions in terms of multiplicative functions. Some consequences of this result are presented.</p>
</abstract>

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.

A Variant of D’Alembert’s Functional Equation on Semigroups with Endomorphisms

Department of Mathematics, Faculty of Sciences

Annales Mathematicae Silesianae

This work is licensed under the Creative Commons Attribution 4.0 International License.

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...

A Variant of D’Alembert’s Functional Equation on Semigroups with Endomorphisms

Published Online: Mar 22, 2022

Page range: 1 - 14

Received: May 01, 2021

Accepted: Feb 24, 2022

DOI: https://doi.org/10.2478/amsil-2022-0004

Keywords
functional equation, d’Alembert, semigroup, multiplicative function, endomorphism

© 2022 Ahmed Akkaoui et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

A Variant of D’Alembert’s Functional Equation on Semigroups with Endomorphisms

Published Online: Mar 22, 2022

Page range: 1 - 14

Received: May 01, 2021

Accepted: Feb 24, 2022

DOI: https://doi.org/10.2478/amsil-2022-0004

Keywordsfunctional equation, d’Alembert, semigroup, multiplicative function, endomorphism

© 2022 Ahmed Akkaoui et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Keywords
functional equation, d’Alembert, semigroup, multiplicative function, endomorphism