On Jensen’s and the quadratic functional equations with involutions

Authors

  • Brahim Fadli Ibn Tofail University.
  • Abdellatif Chahbi Ibn Tofail University.
  • Iz-Iddine El-Fassi Ibn Tofail University.
  • Samir Kabbaj Ibn Tofail University.

DOI:

https://doi.org/10.4067/S0716-09172016000200006

Keywords:

Functional equation, Jensen, quadratic, additive function, semigroup, ecuación funcional, cuadrática, función aditiva, semigrupo.

Abstract

We determine the Solutions f : S → H of the generalized Jensen’s functional equation

f( x + σ(y)) + f( x + τ(y)) = 2f(x), x , y∈ S

and the solutions f : S → H of the generalized quadratic functional equation

f ( x + σ(y)) + f (x + τ(y)) = 2f (x) + 2f (y),    x, y ∈ S,

where S is a commutative semigroup, H is an abelian group (2-torsion free in the first equation and uniquely 2-divisible in the second) and σ, τ are two involutions of S.

Author Biographies

Brahim Fadli, Ibn Tofail University.

Department of Mathematics, Faculty of Sciences.

Abdellatif Chahbi, Ibn Tofail University.

Department of Mathematics, Faculty of Sciences.

Iz-Iddine El-Fassi, Ibn Tofail University.

Department of Mathematics, Faculty of Sciences.

Samir Kabbaj, Ibn Tofail University.

Department of Mathematics, Faculty of Sciences.

References

[1] J. Aczel and J. Dhombres, Functional equations in several variables, Cambridge University Press, New York (1989).

[2] A. Chahbi, B. Fadli, S. Kabbaj, A generalization of the symmetrized multiplicative Cauchy equation, Acta Math. Hungar., pp. 1-7, (2016).

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

[4] B. Fadli, D. Zeglami and S. Kabbaj, On a Gajda’s type quadratic equation on a locally compact abelian group, Indagationes Math., 26, pp. 660-668, (2015).

[5] B. Fadli, D. Zeglami and S. Kabbaj, A variant of Jensen’s functional equation on semigroups, Demonstratio Math., to appear.

[6] P. de Place Friis and H. Stetkær, On the quadratic functional equation on groups, Publ. Math. Debrecen 69, pp. 65-93, (2006).

[7] S-M. Jung, Quadratic functional equations of Pexider type, J. Math. & Math. Sci., 24, pp. 351-359, (2000).

[8] P. Kannappan, Quadratic functional equation and inner product spaces, Results Math., 27, pp. 368-372, (1995).

[9] P. Kannappan, Functional equations and inequalities with applications, Springer, New York, (2009).

[11] C. T. Ng, Jensen’s functional equation on groups, II, Aequationes Math. 58, pp. 311-320, (1999).

[12] C. T. Ng, Jensen’s functional equation on groups, III, Aequationes Math. 62, pp. 143-159, (2001).

[13] C. T. Ng, A Pexider-Jensen functional equation on groups, Aequationes Math. 70, pp. 131-153, (2005).

[14] J. C. Parnami and H.L. Vasudeva, On Jensen’s functional equation, Aequationes Math. 43, pp. 211-218, (1992).

[15] Th. M. Rassias, Inner Product Spaces and Applications, Pitman Research Notes in Mathematics Series, Addison Wesley Longman Ltd,
376, (1997).

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

[17] H. Stetkær, Functional equations on abelian groups with involution, Aequationes Math. 54, pp. 144-172, (1997).

[18] H. Stetkær, On Jensen’s functional equation on groups, Aequationes Math. 66, pp. 100-118, (2003).

[19] H. Stetkær, Functional Equations on Groups, World Scientific Publishing Co, Singapore, (2013).

Published

2017-03-23

How to Cite

[1]
B. Fadli, A. Chahbi, I.-I. El-Fassi, and S. Kabbaj, “On Jensen’s and the quadratic functional equations with involutions”, Proyecciones (Antofagasta, On line), vol. 35, no. 2, pp. 213-223, Mar. 2017.

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