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

Ahmed Redouani,Elhoucien Elqorachi

doi:10.1515/amsil-2017-0006

Abstract

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.

Highlights

Throughout this paper S denotes a semigroup: a set equipped with an associative operation
The first purpose of this paper is to extend the results of Stetkær [33, 34] on the Kannappan’s functional equation (1.4) and Van Vleck’s functional equation (1.5) to the case, where σ is an involutive automorphism of S
By using similar methods and computations to those in [13] we prove that the solutions of (1.4) and (1.5) are closely related to the solutions of the d’Alembert’s classic functional equation (1.2) which has not been studied much on non-abelian semigroups

Summary

Introduction

Throughout this paper S denotes a semigroup: a set equipped with an associative operation. By using similar methods and computations to those in [13] we prove that the solutions of (1.4) and (1.5) are closely related to the solutions of the d’Alembert’s classic functional equation (1.2) (with σ an involutive automorphism) which has not been studied much on non-abelian semigroups. We study the complex-valued solutions of the functional equation (1.4), where σ is an involutive automorphism and μ is a linear combination of Dirac measures (δzi)i∈I , such that zi is in the center of S for all i ∈ I. The non-zero central solutions of the integral Kannappan’s functional equation (1.4), where σ is an involutive automorphism of S, are the functions of the form χ(x) + χ(σ(x)). G is a non-zero solution of d’Alembert’s functional equation (1.2) and satisfies the condition (2.3).

Using this in we obtain

SS S

That is