Note on an Iterative Functional Equation - Baron Karol; Morawiec Janusz (Annales Mathematicae Silesianae, 2024)
We study the problem of solvability of the equation ϕ(x)=∫Ωg(w)ϕ(f(x,ω))P(dω)+F(x),φ \left( x \right) = \int_Ω {g\left( w \right)} φ \left( {f\left( {x,ω } \right)} \right)P\left( {dω } \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.