Let G be a locally compact abelian Hausdorff group, let $$\sigma $$ σ be a continuous involution on G, and let $$\mu ,\nu $$ μ,ν be regular, compactly supported, complex-valued Borel measures on G. We determine the continuous solutions $$f,g:G\rightarrow {\mathbb {C}}$$ f,g:G→C of each of the two functional equations $$\begin{aligned}&\int _{G}f(x+y+t)d\mu (t)+\int _{G}f(x+\sigma (y)+t)d\nu (t)=f(x)g(y),\quad x,y\in G,\\&\int _{G}f(x+y+t)d\mu (t)+\int _{G}f(x+\sigma (y)+t)d\nu (t)=g(x)f(y),\quad x,y\in G, \end{aligned}$$ ∫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.