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Notions as limit sets, invariant or attractors sets are defined in terms of set-valued analysis and relations with different concepts of stable set (in the Lyapunov sense) are established. In this brief paper, we present applications of set-valued analysis to the dynamical system theory and provide a foundational theory in terms of stability analysis.<\/jats:p>","DOI":"10.1007\/s10957-026-02931-5","type":"journal-article","created":{"date-parts":[[2026,1,19]],"date-time":"2026-01-19T19:41:36Z","timestamp":1768851696000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["An underlying theory of multifunctions in discrete-time set-valued dynamical systems"],"prefix":"10.1007","volume":"208","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0447-5620","authenticated-orcid":false,"given":"Elvira","family":"Hern\u00e1ndez","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5496-7362","authenticated-orcid":false,"given":"Juan","family":"Per\u00e1n","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2026,1,19]]},"reference":[{"key":"2931_CR1","unstructured":"Akin, E.: The General Topology of Dynamical Systems. 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