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Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.37 no.2 Antofagasta June 2018

http://dx.doi.org/10.4067/S0716-09172018000200317 

Articles

Solutions and stability of a variant of Wilson's functional equation

Elhoucien Elqorachi1 

Ahmed Redouani2 

1Ibn Zohr University, Faculty of Sciences, Department of Mathematics, Agadir, Morocco. E-mail : e.elqorachi@uiz.ac.ma

2Ibn Zohr University, Faculty of Sciences, Department of Mathematics, Agadir, Morocco. E-mail : Redouani_ahmed@yahoo.fr

Abstract:

In this paper we will investigate the complex-valued solutions and stability of the generalized variant of Wilson's functional equation

(E): 𝑓(xy)+χ(y)𝑓(σ(y)x)=2𝑓(x)g(y), x,y ∈ G,

where G is a group, σ is an involutive morphism of G and χ is a character of G. (a) We solve (E) when σ is an involutive automorphism, and we obtain some properties about solutions of (E) when σ is an involutive anti-automorphism. (b) We obtain the Hyers Ulam stability of equation (E). As an application, we prove the superstability of the functional equation (xy)+χ(y) 𝑓(σ(y)x)=2 𝑓(x) 𝑓(y), x,y ∈ G.

Keywords:  Semigroup-Involution; D'Alembert's equation; Wilson's equation; Automorphism; Homomorphism; Multiplicative function; Hyers-Ulam stability; Superstability

Subjclass: Primary 39B82; Secondary 39B32, 39B52.

Texto completo disponible sólo en PDF.

Full text available only in PDF format.

Acknowledgements.

The authors thank very much the referees for their helpful comments. We would also like to thank the professor H. Stetkær. His help for the final version of this paper is greatly appreciated.

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Received: May 2017; Accepted: April 2018

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