## Abstract

Under the framework given by a growth condition, a Lyapunov property and some continuity assumptions, the present work shows the existence of lower semicontinuous solutions to the Shapley equation for zero-sum semi-Markov games with Borel spaces, weakly continuous transition probabilities and possible unbounded payoff. It is also shown the existence of stationary optimal strategies for the minimizing player and stationary ε-optimal strategies for the maximizing player. These results are proved using a fixed-point approach. Moreover, it is shown the existence of a deterministic stationary minimax strategy for a minimax semi-Markov inventory problem under mild assumptions on the demand distribution.

Original language | English |
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Article number | 9 |

Journal | Acta Applicandae Mathematicae |

Volume | 177 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2022 |

Externally published | Yes |

### Bibliographical note

Funding Information:Work partially supported by Consejo Nacional de Ciencia y Tecnología (CONACYT-Mexico) under grant Ciencia Frontera 2019-87787.

Publisher Copyright:

© 2022, The Author(s), under exclusive licence to Springer Nature B.V.

## Keywords

- Average payoff
- Fixed-point approach
- Lyapunov conditions
- Semi-Markov games
- Shapley equation