Solutions and stability of generalized Kannappan’s and Van Vleck’s functional equations

Elhoucien Elqorachi; Ahmed Redouani

Published: 2017-08-05

Vol. 32 (2018)

Solutions and stability of generalized Kannappan’s and Van Vleck’s functional equations

Elhoucien Elqorachi , Ahmed Redouani

Abstract

We study the solutions of the integral Kannappan’s and Van Vleck’s functional equations
∫_Sf(xyt)dμ(t) + ∫_Sf(xσ(y)t)dμ(t) = 2f(x)f(y), x,y∈S;
∫_Sf(xσ(y)t)dμ(t) - ∫_Sf(xyt)dμ(t) = 2f(x)f(y), x,y∈S,
where S is a semigroup, is an involutive automorphism of S and μ is a linear combination of Dirac measures (δ_{z_i})_i∈I, such that for all i∈I, z_i is in the center of S. We show that the solutions of these equations are closely related to the solutions of the d’Alembert’s classic functional equation with an involutive automorphism. Furthermore, we obtain the superstability theorems for these functional equations in the general case, where σ is an involutive morphism.

Keywords:

Hyers–Ulam stability , semigroup , d’Alembert’s equation , Van Vleck’s equation , Kannappan’s equation , involution , automorphism , multiplicative function , complex measure

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Elqorachi, E., & Redouani, A. (2017). Solutions and stability of generalized Kannappan’s and Van Vleck’s functional equations. Annales Mathematicae Silesianae, 32, 169–200. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/13919

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Elhoucien Elqorachi

Eqorachi@hotmail.com

Affiliation:

Department of Mathematics, Faculty of Sciences, University Ibn Zohr, Morocco Morocco

Ahmed Redouani

Affiliation:

Department of Mathematics, Faculty of Sciences, University Ibn Zohr, Morocco Morocco

Vol. 32 (2018)
Published: 2018-08-24

ISSN: 0860-2107

eISSN: 2391-4238

10.1515/amsil

Publisher

University of Silesia Press

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