SciELO - Scientific Electronic Library Online

vol.35 número2Closed models, strongly connected components and Euler graphsVertex equitable labeling of union of cyclic snake related graphs índice de autoresíndice de materiabúsqueda de artículos
Home Pagelista alfabética de revistas  

Servicios Personalizados




Links relacionados

  • En proceso de indezaciónCitado por Google
  • No hay articulos similaresSimilares en SciELO
  • En proceso de indezaciónSimilares en Google


Proyecciones (Antofagasta)

versión impresa ISSN 0716-0917

Proyecciones (Antofagasta) vol.35 no.2 Antofagasta jun. 2016 

Proyecciones Journal of Mathematics Vol. 35, No 2, pp. 159-176, June 2016. Universidad Católica del Norte Antofagasta - Chile

A generalization of Drygas functional equation

A. Charifi M. Almahalebi 

S. Kabbaj 

Ibn Tofail University



We obtain the Solutions of the following Drygas functional equation

λ ∈Φ f (x + λy + aλ ) = κf(x)+ ∑ λ ∈Φ f(λy), x, y ∈ S

where S is an abelian semigroup, G is an abelian group, f ∈ GS, Φ is a finite automorphism group of S with order k, and aλ ∈ S, λ∈Φ.

Subjclass 2010 Mathematics Subject Classification (MSC) :

20B25; 39B52; 47B39; 65Q20.

Keywords and Phrases : Automorphism group; difference opera-tor; Drygas functional equation.


[1]    M. Ait Sibaha, B. Bouikhalene, E. Elqorachi, Hyers-Ulam-Rassias sta-bility of the K-quadratic functional equation, J. Ineq. Pure and appl. Math 8, (2007).

[2]    L. M. Arriola, W. A. Beyer, Stability ofthe Cauchy functional equation over p-adic fields, Real Analysis Exchange, 31 (1), pp. 125-132, (2005).

[3]    J. Baker, A general functional equation and its stability, Proceedings of the American Mathematical Society, 133(6), pp. 1657-1664, (2005).

[4]    B. Bouikhalene and E. Elqorachi, Hyers-Ulam-Rassias stability ofthe Cauchy linear functional equation, Tamsui Oxford Journal of Mathe-matical Sciences 23 (4), pp. 449-459, (2007).

[5]    A. Charifi, B. Bouikhalene, E. Elqorachi, Hyers-Ulam-Rassias stability of a generalized Pexider functional equation, Banach J. Math. Anal, 1 (2), pp. 176-185, (2007).

[6]    A. Charifi,B. Bouikhalene, E. Elqorachi, A. Redouani, Hyers-Ulam-Rassias stability of a generalized Jensen functional equation, Aust. J. Math. Anal. Appl, 6 (1), pp. 1-16, (2009).

[7]    AB. Chahbi, A. Charifi, B. Bouikhalene, S. Kabbaj, Operatorial ap-proach to the non-Archimedean stability of a Pexider K-quadratic functional equation, Arab Journal of Mathematical Sciences, 21 (1), pp. 67-83, (2015).

[8]    D. Z. Djokovic, A representation theorem for (X\ — 1)(X2 — 1)...(Xn — 1) and its applications, In Annales Polonici Mathematici 22 (2), pp. 189198, (1969).

[9]    H. Drygas, Quasi-inner producís and their applications, Springer Netherlands., pp. 13-30, (1987).

[10] B. R. Ebanks, P. L. Kannappan, P. K. Sahoo, A common generalization of functional equations characterizing normed and quasi-inner-product spaces, Canad. Math. Bull, 35 (3), pp. 321-327, (1992).

[11]    V. A. Faiziev, P. K. Sahoo, On Drygas functional equation on groups, Int. J. Appl. Math. Stat. 7, pp. 59-69, (2007).

[12]    M. Frechet, Une définition fonctionnelles des polynómes, Nouv. Ann. 9, pp. 145-162, (1909).

[13]    A. Gianyi, A characterization of monomial functions, Aequationes Math. 54, pp. 343-361, (1997).

[14]    D. H. Hyers, Transformations with bounded n-th differences, Pacific J. Math., 11, pp. 591-602, (1961).

[15]    S. -M. Jung, Stability of the quadratic equation of Pexider type, Abh. Math. Sem. Univ. Hamburg, 70, pp. 175-190, (2000).

[16]    S. -M. Jung, P. K. Sahoo, Hyers-Ulam stability of the quadratic equation of Pexider type, J. Korean Math. Soc., 38 (3), pp. 645-656, (2001).

[17]    S.-M. Jung, P. K. Sahoo, Stability of a functional equation of Drygas, Aequationes Math., 64 (3), pp. 263-273, (2002).

[18]    R. Lukasik, Some generalization of Cauchy’s and the quadratic functional equations, Aequat. Math., 83, pp. 75-86, (2012).

[19]    S. Mazur, W. Orlicz, Grundlegende Eigenschaften der Polynomischen Operationen, Erst Mitteilung, Studia Math., 5, pp. 50-68, (1934).

[20]    A. K. Mirmostafaee, Non-Archimedean stability of quadratic equations, Fixed Point Theory, 11 (1), pp. 67-75, (2010).

[21]    P. K. Sahoo and Pl. Kannappan, Introduction to Functional Equations, CRC Press, Boca Raton, Florida, (2011).

[22]    P. Sinopoulos, Functional equations on semigroups, Aequationes Math. 59, pp.255-261, (2000).

[23]    W. Smajdor, On set-valued solutions of a functional equation of Drygas, Aequ. Math. 77, pp. 89-97, (2009).

[24]    H. Stetkffir, Functional equations on abelian groups with involution. II, Aequationes Math. 55, pp. 227-240, (1998).

[25]    H. Stetkffir, Functional equations involving means of functions on the complex plane, Aequationes Math. 55, pp. 47-62, (1998).

[26]    Gy. Szabo, Some functional equations related to quadratic functions, Glasnik Math. 38, pp. 107-118, (1983).

[27]    D. Yang, Remarks on the stability of Drygas equation and the Pexider-quadratic equation, Aequationes Math. 68, pp. 108-116, (2004).

A. Charifi

Department of Mathematics, Faculty of Sciences,

Ibn Tofail University,

BP: 133, 14000,



e-mail :

M. Almahalebi

Department of Mathematics, Faculty of Sciences,

Ibn Tofail University,

BP: 133, 14000,



e-mail :

S. Kabbaj

Department of Mathematics, Faculty of Sciences,

Ibn Tofail University,

BP: 133, 14000,



e-mail :

Received : April 2015. Accepted : April 2016

Creative Commons License Todo el contenido de esta revista, excepto dónde está identificado, está bajo una Licencia Creative Commons