Lipschitzian solutions to inhomogeneous linear iterative equations

We study the problems of the existence, uniqueness and continuous dependence of Lipschitzian solutions $\phi$ of equations of the form $$\phi (x)={\int }_{\mathrm{\Omega }}g(\omega )\phi (f(x,\omega ))\mu (d\omega )+F(x),$$ where $\mu$ is a measure on a $\sigma$-algebra of subsets of a set $\mathrm{\Omega }$.