<abstract xmlns="http://www.w3.org/1999/xhtml"> 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, σ is an involutive automorphism of S and µ is a linear combination of Dirac measures ( ᵟ zi)I ∈ I, such that for all i ∈ I, ziis 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.</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, σ is an involutive automorphism of S and µ is a linear combination of Dirac measures ( ᵟ zi)I ∈ I, such that for all i ∈ I, ziis 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.

Solutions and Stability of Generalized Kannappan’s and Van Vleck’s Functional Equations

Annales Mathematicae Silesianae

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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;...

Solutions and Stability of Generalized Kannappan’s and Van Vleck’s Functional Equations

Published Online: Aug 24, 2018

Page range: 169 - 200

Received: Nov 17, 2016

Accepted: May 03, 2017

DOI: https://doi.org/10.1515/amsil-2017-0006

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

© 2018 Elhoucien Elqorachi, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Solutions and Stability of Generalized Kannappan’s and Van Vleck’s Functional Equations

Published Online: Aug 24, 2018

Page range: 169 - 200

Received: Nov 17, 2016

Accepted: May 03, 2017

DOI: https://doi.org/10.1515/amsil-2017-0006

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

© 2018 Elhoucien Elqorachi, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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