Solution of several functional equations on abelian groups with involution

Let G be a locally compact abelian Hausdorff group, let σ be a continuous involution on G, and let μ, ν be regular, compactly supported, complex-valued Borel measures on G. We determine the continuous solutions f, g: G→ C of each of the two functional equations ∫Gf(x+y+t)dμ(t)+∫Gf(x+σ(y)+t)dν(t)=f(x)g(y),x,y∈G,∫Gf(x+y+t)dμ(t)+∫Gf(x+σ(y)+t)dν(t)=g(x)f(y),x,y∈G,in terms of characters and additive functions. These equations provides a common generalization of many functional equations such as d’Alembert’s, Cauchy’s, Gajda’s, Kannappan’s, Van Vleck’s, or Wilson’s equations. So, several functional equations will be solved. © 2017, African Mathematical Union and Springer-Verlag GmbH Deutschland.