<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

Data publikacji: 29 sie 2023

Zakres stron: 169 - 184

Otrzymano: 20 paź 2022

Przyjęty: 10 sie 2023

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

Słowa kluczowe
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

Data publikacji: 29 sie 2023

Zakres stron: 169 - 184

Otrzymano: 20 paź 2022

Przyjęty: 10 sie 2023

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

Słowa kluczowePexider, 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.

Słowa kluczowe
Pexider, Drygas functional equation, quadratic, symmetric bi-additive function, additive function, semigroup, endomorphism