Integral Van Vleck's and Kannappan's functional equations on semigroups

2016/05/12 by Elqorachi Elhoucien, Elhoucien, Elqorachi
Mathematics · #39B32 #39B52 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA #msc:39B32 #msc:39B52

paper · pdf · doi:10.48550/arxiv.1605.05248

12pages

arxiv created 2016/05/12 · arxiv updated 2016/05/18

Abstract

In this paper we study the solutions of the integral Van Vleck's functional equation for the sine ∫_Sf(xτ(y)t)dμ(t)-∫_Sf(xyt)dμ(t) =2f(x)f(y), x,y∈ S and the integral Kannappan's functional equation ∫_Sf(xyt)dμ(t)+∫_Sf(xτ(y)t)dμ(t) =2f(x)f(y), x,y∈ S, where S is a semigroup, τ is an involution of S and μ is a measure that is linear combinations of point measures (δ_z_i)_{i∈ I}, such that for all i∈ I, z_i is contained in the center of S. We express the solutions of the first equation by means of multiplicative functions on S, and we prove that the solutions of the second equation are closely related to the solutions of the classic d'Alembert's functional equation with involution.

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