On the existence of relative values for undiscounted multichain Markov decision processes

The coupled functional equations of undiscounted multichain semi-Markovian decision processes are shown to possess a solution by converting the value equation into the form $\text{v =}\text{max}\text{w(?) + Π(?)v; ? ? S] ≡ Qv}$ where S is the set of maximal-gain policies and $\text{w(?)}$ is the bias vector associated with policy ?. An elementary proof then shows that the operator Q possesses a fixed point.