Stochastic fixed points involving the maximum

We consider distributional fixed point equations of the form X D = ∨iTiXi in a systematic way. The distribution of T = (T1, T2, . . .) is given in advance. The positive rvs T,Xi, i ∈ IN are independent and the Xi have the same distribution as X. We present a systematic approach in order to find solutions using the monotonicity of the corresponding operator. These equations in the limit come up in the natural setting of trees with finite or countable many branches. Examples are in branching processes and the analysis of algorithms (for parallel computing).