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

<p>Let (<italic>S, </italic>+) be an abelian semigroup, let (<italic>H, </italic>+) be an abelian group which is uniquely 2-divisible, and let <italic>ϕ </italic>be an endomorphism of <italic>S</italic>. We find the solutions <italic>f, h </italic>: <italic>S </italic>→ <italic>H </italic>of each of the functional equations
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2023-0015_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>ϕ</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>=</mo><mi>h</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mo>∘</mo><mi>ϕ</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>,</mo><mi> </mi><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>S</mi><mo>,</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>ϕ</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>=</mo><mi>h</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><mn>2</mn><mi>f</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>,</mo><mi> </mi><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>S</mi><mo>,</mo></mrow></mtd></mtr></mtable></mrow>
</math>
<tex-math>\matrix{   {f\left( {x + y} \right) + f\left( {x + \varphi \left( y \right)} \right) = h\left( x \right) + f\left( y \right) + f \circ \varphi \left( y \right),\,x,y \in S,} \hfill  \cr    {f\left( {x + y} \right) + f\left( {x + \varphi \left( y \right)} \right) = h\left( x \right) + 2f\left( y \right),\,x,y \in S,} \hfill  \cr  }</tex-math>
</alternatives>
</disp-formula>
in terms of additive and bi-additive maps. Moreover, as applications, we determine the solutions of some related functional equations.</p>
</abstract>

Let (S, +) be an abelian semigroup, let (H, +) be an abelian group which is uniquely 2-divisible, and let ϕ be an endomorphism of S. We find the solutions f, h : S → H of each of the functional equations



f(x+y)+f(x+ϕ(y))=h(x)+f(y)+f∘ϕ(y), x,y∈S,f(x+y)+f(x+ϕ(y))=h(x)+2f(y), x,y∈S,

\matrix{   {f\left( {x + y} \right) + f\left( {x + \varphi \left( y \right)} \right) = h\left( x \right) + f\left( y \right) + f \circ \varphi \left( y \right),\,x,y \in S,} \hfill  \cr    {f\left( {x + y} \right) + f\left( {x + \varphi \left( y \right)} \right) = h\left( x \right) + 2f\left( y \right),\,x,y \in S,} \hfill  \cr  }


in terms of additive and bi-additive maps. Moreover, as applications, we determine the solutions of some related functional equations.

New Pexiderizations of Drygas’ Functional Equation on Abelian Semigroups

Annales Mathematicae Silesianae

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

{"article-title":"New Pexiderizations of Drygas\u2019 Functional Equation on Abelian Semigroups"}

Let (S, +) be an abelian semigroup, let (H, +) be an abelian group which is uniquely 2-divisible, and let ϕ be an endomorphism of S. We find the solutions f, h...

New Pexiderizations of Drygas’ Functional Equation on Abelian Semigroups

Publicado en línea: 29 ago 2023

Páginas: 169 - 184

Recibido: 20 oct 2022

Aceptado: 10 ago 2023

DOI: https://doi.org/10.2478/amsil-2023-0015

Palabras clave
Pexider, Drygas functional equation, quadratic, symmetric bi-additive function, additive function, semigroup, endomorphism

© 2023 Youssef Aissi et al., published by Sciendo

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

New Pexiderizations of Drygas’ Functional Equation on Abelian Semigroups

Publicado en línea: 29 ago 2023

Páginas: 169 - 184

Recibido: 20 oct 2022

Aceptado: 10 ago 2023

DOI: https://doi.org/10.2478/amsil-2023-0015

Palabras clavePexider, Drygas functional equation, quadratic, symmetric bi-additive function, additive function, semigroup, endomorphism

© 2023 Youssef Aissi et al., published by Sciendo

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

Palabras clave
Pexider, Drygas functional equation, quadratic, symmetric bi-additive function, additive function, semigroup, endomorphism