Note on an Iterative Functional Equation

We study the problem of solvability of the equation ϕ(x)=∫Ωg(w)ϕ(f(x,ω))P(dω)+F(x), \varphi \left( x \right) = \int_\Omega {g\left( w \right)} \varphi \left( {f\left( {x,\omega } \right)} \right)P\left( {d\omega } \right) + F\left( x \right), where P is a probability measure on a σ-algebra of subsets of Ω, assuming Hölder continuity of F on the range of f.