Let (Ω,𝓐,P) be a probability space and let τ: ℝ × Ω → ℝ be a mapping strictly increasing and continuous with respect to the first variable, and 𝓐-measurable with respect to the second variable. We discuss the problem of existence of probability distribution solutions of the general linear equation
$F(x) = ∫_Ω F(τ(x,ω)) P(dω)$.
We extend our uniqueness-type theorems obtained in Ann. Polon. Math. 95 (2009), 103-114.