<abstract xmlns="http://www.w3.org/1999/xhtml"><p>Let (<italic>S</italic>, +) be a commutative semigroup, σ : <italic>S</italic> → <italic>S</italic> be an endomorphism with σ<sup>2</sup> = <italic>id</italic> and let <italic>K</italic> be a field of characteristic different from 2. Inspired by the problem of strong alienation of the Jensen equation and the exponential Cauchy equation, we study the solutions <italic>f</italic>, <italic>g</italic> : <italic>S</italic> → <italic>K</italic> of the functional equation
<disp-formula id="j_amsil-2016-0007_eq_001_w2aab2b8ab1b7b1aab1c14b1c21Aa"><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2016-0007_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo stretchy="false">)</mo><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mtext>for</mtext><mi> </mi><mi> </mi><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>S</mi><mo>.</mo></math><tex-math>$$f(x + y) + f(x + \sigma (y)) + g(x + y) = 2f(x) + g(x)g(y)\;\;\;\;{\rm for}\;\;x,y \in S.$$</tex-math></alternatives></disp-formula>
We also consider an analogous problem for the Jensen and the d’Alembert equations as well as for the d’Alembert and the exponential Cauchy equations.</p></abstract>

Let (S, +) be a commutative semigroup, σ : S → S be an endomorphism with σ2 = id and let K be a field of characteristic different from 2. Inspired by the problem of strong alienation of the Jensen equation and the exponential Cauchy equation, we study the solutions f, g : S → K of the functional equation
f(x+y)+f(x+σ(y))+g(x+y)=2f(x)+g(x)g(y)     for  x,y∈S.$$f(x + y) + f(x + \sigma (y)) + g(x + y) = 2f(x) + g(x)g(y)\;\;\;\;{\rm for}\;\;x,y \in S.$$
We also consider an analogous problem for the Jensen and the d’Alembert equations as well as for the d’Alembert and the exponential Cauchy equations.

Alienation of the Jensen, Cauchy and d’Alembert Equations

Faculty of Mathematics and Natural Sciences, University of Rzeszów, Pigonia 1, 35-310 Rzeszów,

Annales Mathematicae Silesianae

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Alienation of the Jensen, Cauchy and d\u2019Alembert Equations"}

Let (S, +) be a commutative semigroup, σ : S → S be an endomorphism with σ2 = id and let K be a field of characteristic different from 2. Inspired by the...

Alienation of the Jensen, Cauchy and d’Alembert Equations

Published Online: Sep 23, 2016

Page range: 181 - 191

Received: Apr 01, 2016

Accepted: May 21, 2016

DOI: https://doi.org/10.1515/amsil-2016-0007

Keywords
Alienation, exponential Cauchy equation, Jensen equation, d’Alembert equation

© 2016 Barbara Sobek, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Alienation of the Jensen, Cauchy and d’Alembert Equations

Published Online: Sep 23, 2016

Page range: 181 - 191

Received: Apr 01, 2016

Accepted: May 21, 2016

DOI: https://doi.org/10.1515/amsil-2016-0007

KeywordsAlienation, exponential Cauchy equation, Jensen equation, d’Alembert equation

© 2016 Barbara Sobek, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Alienation, exponential Cauchy equation, Jensen equation, d’Alembert equation