## Articulo

• Similares en SciELO
• Similares en Google

## versión impresa ISSN 0716-0917

### Proyecciones (Antofagasta) vol.38 no.2 Antofagasta jun. 2019

#### http://dx.doi.org/10.4067/S0716-09172019000200221

Articles

A sine type functional equation on a topological group

1Moulay Ismail University, Department of Mathematics, E.N.S.A.M, B. P: 15290 Al Mansour, Meknes, Morocco. e-mail : zeglamidriss@yahoo.fr

2Ibn Tofail University, Department of Mathematics, Faculty of Sciences, B. P. : 14000. Kenitra, Morocco. e-mail : samkabbaj@yahoo.fr

3Ibn Tofail University, Department of Mathematics, Faculty of Sciences, B. P. : 14000. Kenitra, Morocco. e-mail : tialmohamed@gmail.com

Abstract

In [13] H. Stetkær obtained the complex valued solutions of the functional equation

f(xyz0)f(xy−1z0) = f(x)2 − f(y)2, x, y ∈ G,

where G is a topological group and z0 ∈ Z(G) (the center of G). Our main goal is first to remove this restriction and second, when G is 2-divisible and abelian, we will investigate the superstability of the above functional equation.

Keywords: Sine functional equation; Character; Additive map; Superstability

Acknowledgement.

The authors wishe to thank the referee for a number of constructive comments which have led to essential improvement of the paper. We are indebted to Professor Henrik Stetkær for a helpful conversation which contributed to the first part of this article.

References

[1] J. Aczél, J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, New York, (1989). [ Links ]

[2] J. A. Baker, J. Lawrence, and F. Zorzitto, The stability of the equation f(x + y) = f(x)f(y), Proc. Amer. Math. Soc. 74, pp. 242-246, (1979). [ Links ]

[3] J. A. Baker, The stability of the cosine equation, Proc. Amer. Math. Soc. 80 , pp. 411-416, (1980). [ Links ]

[4] P. W. Cholewa, The stability of the sine equation, Proc. Amer. Math. Soc. 88, No. 4, pp. 631-634, (1983). [ Links ]

[5] J. Chung and D. Kim, Sine functional equation in several variables, Archiv der Mathematik 86, No 5, pp. 425-429, (2006). [ Links ]

[6] I. Corovei, The sine functional equation on 2-divisible groups, Mathematica 47, No 1, pp. 49-52, (2005). [ Links ]

[7] Pl. Kannappan, On sine functional equation, Studia Sci. Math. Hung., 4, pp. 331-333, (1969). [ Links ]

[8] Pl. Kannappan, Functional Equations and Inequalities with Applications. Springer Monographs in Mathematics. Springer, New York, xxiv+810, (2009). [ Links ]

[9] G. H. Kim, A stability of the generalized sine functional equations, J. Math. Anal. Appl., 331, pp. 886-894, (2007). [ Links ]

[10] S. Kurepa, On the functional equation f(x+y)f(x−y) = f(x)2−f(y)2, Ann. Polon. Math., 10 , pp. 1-5, (1961). [ Links ]

[11] A. Roukbi, D. Zeglami and S. Kabbaj, Hyers-Ulam stability of Wilson’s functional equation, J. Math. Sci. Adv. Appl., 22, pp. 19-26, (2013). [ Links ]

[12] P. Sinopoulos, Generalized sine equations, I, Aequationes math., 48, No. 2, pp. 171-193, (1994). [ Links ]

[13] H. Stetkær, Functional equations on groups, World Scientific Publishing, Hackensack, xvi+378, (2013). [ Links ]

[14] H. Stetkær , Van Vleck’s functional equation for the sine. Aequationes Math. 90 (1), pp. 25-34, (2016). [ Links ]

[15] E. B. Van Vleck, A functional equation for the sine, Ann. of Math., Second Series, 11 (4), pp. 161-165, (1910). [ Links ]

[16] D. Zeglami, B. Fadli, S. Kabbaj , On a variant of µ-Wilson’s functional equation on a locally compact group, Aequationes Math., 89, pp. 1265-1280, (2015). [ Links ]

[17] D. Zeglami , A. Charifi andS. Kabbaj , Superstability problem for a large class of functional equations, Afr. Mat., 27, pp. 469-484, (2016). [ Links ]

[18] D. Zeglami , M. Tial andB. Fadli , Wilson’s Type Hilbert-space valued functional equations, Adv. Pure. Appl. Math., 7, No. 3, pp. 189-196, (2016). [ Links ]

[19] D. Zeglami andB. Fadli , Integral functional equations on locally compact groups with involution, Aequationes Math. , 90 (5), pp. 967-982. [ Links ]

[20] D. Zeglami , M. Tial, S. Kabbaj , The integral sine addition law, Submited to Proyecciones J. of Math. [ Links ]

Received: April 2017; Accepted: March 2019

This is an open-access article distributed under the terms of the Creative Commons Attribution License