On the second optimality equation for semi-Markov decision models

For a semi-Markov decision model with average return, the validity of the second optimality equation is shown in the (nonmodified) form where the actions run through the set of all admissible actions rather than through the set of maximum points (conserving actions) for the first optimality equation. As a consequence the existence of a strongly optimal stationary policy is shown. The results seem to be known only for finite state finite action models whereas here countable state compact action models with unbounded rewards are considered.