Solutions and stability of a variant of Van Vleck's and d'Alembert's functional equations

2016/08/09 by Elhoucien Elqorachi, Elhoucien, Elqorachi, Ahmed Redouani +3
Mathematics · #39B32 #39B52 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Equations Stability Results #Mathematical and Theoretical Analysis

paper · pdf · doi:10.48550/arxiv.1608.03906

openalex publication_date 2016/08/09 · openalex created_date 2016/09/16 · openalex updated_date 2026/07/28

Abstract

In this paper. (1) We determine the complex-valued solutions of the following variant of Van Vleck's functional equation ∫_Sf(σ(y)xt)dμ(t)-∫_Sf(xyt)dμ(t) = 2f(x)f(y), x,y∈ S, where S is a semigroup, σ is an involutive morphism of S, and μ is a complex measure that is linear combinations of Dirac measures (δ_z_i)_{i∈ I}, such that for all i∈ I, z_i is contained in the center of S. (2) We determine the complex-valued continuous solutions of the following variant of d'Alembert's functional equation ∫_Sf(xty)dυ(t)+∫_Sf(σ(y)tx)dυ(t) = 2f(x)f(y), x,y∈ S, where S is a topological semigroup, σ is a continuous involutive automorphism of S, and υ is a complex measure with compact support and which is σ-invariant. (3) We prove the superstability theorems of the first functional equation.

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