Lipschitzian solutions to inhomogeneous linear iterative equations

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