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# Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia

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Volume 68, Issue 2

# A special class of functional equations

Ahmed Charifi
/ Driss Zeglami
Published Online: 2018-03-31 | DOI: https://doi.org/10.1515/ms-2017-0110

## Abstract

We obtain in terms of additive and multi-additive functions the solutions f, h: SH of the functional equation

$∑λ∈Φf(x+λy+aλ)=Nf(x)+h(y),x,y∈S,$

where (S, +) is an abelian monoid, Φ is a finite group of automorphisms of S, N = | Φ | designates the number of its elements, {aλ, λ ∈ Φ} are arbitrary elements of S and (H, +) is an abelian group. In addition, some applications are given. This equation provides a joint generalization of many functional equations such as Cauchy’s, Jensen’s, Łukasik’s, quadratic or Φ-quadratic equations.

## References

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Charifi, A.—Zeglami, D.—Kabbaj, S.: Solution of generalized Jensen and quadratic functional equations, submited.Google Scholar

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Chung, J. K.—Ebanks, B. R.—Ng, C. T.—Sahoo, P. K.: On a quadratic trigonometric functional equation and some applications, Trans. Amer. Math. Soc. 347 (1995), 1131–1161.

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Czerwik, S.: On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64.

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Ebanks, B. R.—Kannappan, PL.—Sahoo, P. K.: A common generalization of functional equations characterizing normed and quasi-inner-product spaces, Canad. Math. Bull. 35 (1992), 321–327.

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Fadli, B.—Zeglami, D.—Kabbaj, S.: On a Gajda’s type quadratic equation on a locally compact abelian group, Indagationes Math. 26 (2015), 660–668.

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Fadli, B.—Zeglami, D.—Kabbaj, S.: A variant of Jensen’s functional equation on semigroups, Demonstratio Math. 49 (2016), 413–420.Google Scholar

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Łukasik, R.: Some generalization of Cauchy’s and the quadratic functional equations, Aequationes Math. 83 (2012), 75–86.

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Stetkæ r, H.: Functional equations on abelian groups with involution, Aequationes Math. 54 (1997), 144–172.

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Accepted: 2016-08-25

Published Online: 2018-03-31

Published in Print: 2018-04-25

Citation Information: Mathematica Slovaca, Volume 68, Issue 2, Pages 397–404, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918,

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