Iterated random functions and regularly varying tails

We consider solutions to so-called stochastic fixed point equation $R \stackrel{d}{=} \Psi(R)$, where $\Psi$ is a random Lipschitz function and $R$ is a random variable independent of $\Psi$. Under the assumption that $\Psi$ can be approximated by the function $x \mapsto Ax+B$ we show that the tail of $R$ is comparable with the one of $A$, provided that the distribution of $\log (A\vee 1)$ is tail equivalent. In particular we obtain new results for the random difference equation.