## Abstract

In the paper some known and new extensions of the famous theorem of Filippov (1967) and a theorem of Pliś (1965) for differential inclusions are presented. We replace the Lipschitz condition on the set-valued map in the right-hand side by a weaker onesided Lipschitz (OSL), one-sided Kamke (OSK) or a continuity-like condition (CLC). We prove new Filippov-type theorems for singularly perturbed and evolution inclusions with OSL right-hand sides. In the CLC case we obtain two extended theorems, one of which implies directly the relaxation theorem. We obtain also a theorem in Banach spaces for OSK multifunctions. Some applications to exponential formulae are surveyed.

Original language | English |
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Pages (from-to) | 1251-1271 |

Number of pages | 21 |

Journal | Control and Cybernetics |

Volume | 38 |

Issue number | 4 |

State | Published - 2009 |

## Keywords

- Differential inclusions
- Filippov
- Kamke
- Lipschitz
- One-sided
- Pliś