Alienation of the Jensen, Cauchy and d’Alembert equations

Barbara Sobek

Published: 2016-09-23

Vol. 30 (2016)

Alienation of the Jensen, Cauchy and d’Alembert equations

Barbara Sobek

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.
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.

Keywords:

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

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Sobek, B. (2016). Alienation of the Jensen, Cauchy and d’Alembert equations. Annales Mathematicae Silesianae, 30, 181–191. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/13962

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Vol. 30 (2016)
Published: 2016-09-23

ISSN: 0860-2107

eISSN: 2391-4238

10.2478/amsil

Publisher

University of Silesia Press

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