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

Download files

PDF

Citation rules

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

Similar Articles

Raghavendra G. Kulkarni, Insert a root to extract a root of quintic quickly , Annales Mathematicae Silesianae: Vol. 33 (2019)
Youssef Aissi, Driss Zeglami, Brahim Fadli, Alienation of Drygas' and Cauchy's functional equations , Annales Mathematicae Silesianae: Vol. 35 No. 2 (2021)
Roman Ger, The alienation phenomenon and associative rational operations , Annales Mathematicae Silesianae: Vol. 27 (2013)
Jens Schwaiger, Connections between the completion of normed spaces over non-archimedean fields and the stability of the Cauchy equation , Annales Mathematicae Silesianae: Vol. 34 No. 1 (2020)
Bruce Ebanks, A Levi–Civita equation on monoids, two ways , Annales Mathematicae Silesianae: Vol. 36 No. 2 (2022)
PL. Kannappan, P.K. Sahoo, J.K. Chung, An equation associated with the distance between probability distributions , Annales Mathematicae Silesianae: Vol. 8 (1994)
Detlef Gronau, Maciej Sablik, A functional equation arising from an asymptotic formula for iterates , Annales Mathematicae Silesianae: Vol. 8 (1994)
Roman Ger, Justyna Sikorska, On the Cauchy equation on spheres , Annales Mathematicae Silesianae: Vol. 11 (1997)
Beata Hejmej, Stability of functional equations in dislocated quasimetric spaces , Annales Mathematicae Silesianae: Vol. 32 (2018)
Roman Badora, Barbara Przebieracz, Peter Volkmann, Stability of the Pexider functional equation , Annales Mathematicae Silesianae: Vol. 24 (2010)

1 2 3 4 5 6 7 8 9 10 > >>

You may also start an advanced similarity search for this article.

References

1. Dhombres J., Relations de dépendance entre les équations fonctionnelles de Cauchy, Aequationes Math. 35 (1988), 186–212.
2. Fechner W., A characterization of quadratic-multiplicative mappings, Monatsh. Math. 164 (2011), 383–392.
3. Ger R., Additivity and exponentiality are alien to each other, Aequationes Math. 80 (2010), 111–118.
4. Ger R., The alienation phenomenon and associative rational operations, Ann. Math. Sil. 27 (2013), 75–88.
5. Ger R., Alienation of additive and logarithmic equations, Ann. Univ. Sci. Budapest. Sect. Comput. 40 (2013), 269–274.
6. Ger R., Reich L., A generalized ring homomorphisms equation, Monatsh. Math. 159 (2000), 225–233.
7.. Kominek Z., Sikorska J., Alienation of the logarithmic and exponential functional equations, Aequationes Math. 90 (2016), 107–121.
8. Maksa Gy., Sablik M., On the alienation of the exponential Cauchy equation and the Hosszú equation, Aequationes Math. 90 (2016), 57–66.
9. Sinopoulos P., Generalized sine equations I, Aequationes Math. 48 (1994), 171–193.
10. Sinopoulos P., Functional equations on semigroups, Aequationes Math. 59 (2000), 255–261. Google Scholar

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

ISSN: 0860-2107

eISSN: 2391-4238

10.1515/amsil

Publisher

University of Silesia Press

Submit