Abstract
We prove a viability result for differential inclusions involving the Stieltjes derivative with respect to a left-continuous non-decreasing function with time dependent state constraints. A tangential condition using a generalized notion of the contingent derivative is imposed. Classical viability results (for usual differential inclusions) are thus generalized and, at the same time, the gate to new viability results for difference inclusions, impulsive differential inclusions or dynamic inclusions on time scales is open. As a consequence, in the particular case where the state constraint is a tube, a Filippov-type lemma is obtained for this very general setting, of differential problems driven by Stieltjes derivatives.
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We would like to thank the two reviewers for all their valuable comments and suggestions which helped us to improve the quality of the manuscript.
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Open access funding provided by HEAL-Link Greece. This work was supported by the project "Interdisciplinary Cloud and Big Data Center at Ștefan cel Mare University of Suceava", POC/398/1/1, 343/390019, co-founded by the European Union.
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Communicated by Minh N. Dao.
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Satco, B., Smyrlis, G. Viability and Filippov-type Lemma for Stieltjes Differential Inclusions. Set-Valued Var. Anal 31, 31 (2023). https://doi.org/10.1007/s11228-023-00690-3
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DOI: https://doi.org/10.1007/s11228-023-00690-3