An extensions of Kannappan's and Van Vleck's functional equations on semigroups

2016/11/16 by Elhoucien, Elqorachi, Ahmed, Redouani
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1611.06861

Abstract

This paper treats two functional equations, the Kannppan-Van Vleck functional equation μ(y)f(xτ(y)z₀)± f(xyz₀) =2f(x)f(y), x,y∈ S and the following variant of it μ(y)f(τ(y)xz₀)± f(xyz₀) = 2f(x)f(y), x,y∈ S, in the setting of semigroups S that need not be abelian or unital, τ is an involutive morphism of S, μ : S\longrightarrow ℂ is a multiplicative function such that μ(xτ(x))=1 for all x∈ S and z₀ is a fixed element in the center of S. We find the complex-valued solutions of these equations in terms of multiplicative functions and solutions of d'Alembert's functional equation.

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