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Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.37 no.3 Antofagasta Sept. 2018

http://dx.doi.org/10.4067/S0716-09172018000300565 

Articles

An integral functional equation on groups under two measures

B. Fadli1 

D. Zeglami2 

S. Kabbaj3 

1IBN Tofail University, Department of Mathematics, Faculty of Sciences, B. P. : 14000. Kenitra, Morocco e-mail : himfadli@gmail.com

2Moulay ISMAIL University, E. N. S. A. M, Department of Mathematics, B. P. : 15290 Al Mansour-MEKNES, Morocco, e-mail : zeglamidriss@yahoo.fr

3IBN Tofail University, Department of Mathematics, Faculty of Sciences, B. P. : 14000. Kenitra, Morocco e-mail : samkabbaj@yahoo.fr

Abstract

Let G be a locally compact Hausdorff group, let σ be a continuous involutive automorphism on G, and let μ, ν be regular, compactly supported, complex-valued Borel measures on G. We find the continuous solutions 𝑓 : G → C of the functional equation

in terms of continuous characters of G. This equation provides a common generalization of many functional equations (d’Alembert’s, Cauchy’s, Gajda’s, Kannappan’s, Stetkær’s, Van Vleck’s equations...). So, a large class of functional equations will be solved.

Keywords: Functional equation; Van Vleck; Kannappan; involutive automorphism; group character.

Texto completo disponible sólo en PDF.

Full text available only in PDF format.

References

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[2] Fadli, B., Zeglami, D., Kabbaj, S.: A joint generalization of Van Vleck’s and Kannappan’s equations on groups. Adv. Pure Appl. Math. 6 (3), pp. 179-188, (2015). [ Links ]

[3] Gajda, Z.: A generalization of d’Alembert’s functional equation, Funkcial. Ekvac. 33 (1), pp. 69-77, (1990). [ Links ]

[4] Kannappan, PL.: A functional equation for the cosine. Can. Math. Bull. 11, pp. 495-498, (1968). [ Links ]

[5] Kannappan, PL.: Functional equations and inequalities with applications. Springer, New York, (2009). [ Links ]

[6] Perkins, A.M., Sahoo, P.K.: On two functional equations with involution on groups related to sine and cosine functions. Aequationes Math . 89 (5), pp. 1251-1263, (2015). [ Links ]

[7] Stetkær, H.: Functional equations on groups. World Scientific, Publishing Co, Singapore, (2013). [ Links ]

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

[9] Stetkær, H.: Kannappan’s functional equation on semigroups with involution. Semigroup Forum. 94 (1), pp. 17-30, (2017). [ Links ]

[10] Van Vleck, E.B.: A functional equation for the sine. Ann. Math. 7, pp. 161-165, (1910). [ Links ]

Accepted: May 2018; Received: November 2017

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