An existence and qualitative result for discontinuous implicit differential equations
LetT > 0 and Y ⊆ R^n. Given a function f:[0,T]×R^n×Y → R,we consider the Cauchy problem f(t,u,u′) = 0 in [0,T], u(0) = ξ . We prove an existence and qualitative result for the generalized solutions of the above problem. In particular, our result does not require the continuity of f with respect t...
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| Format: | Article |
| Language: | English |
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Accademia Peloritana dei Pericolanti
2018-10-01
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| Series: | Atti della Accademia Peloritana dei Pericolanti : Classe di Scienze Fisiche, Matematiche e Naturali |
| Online Access: |
http://dx.doi.org/10.1478/AAPP.962A4
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| _version_ | 1811323515639431168 |
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| author | Paolo Cubiotti |
| author_facet | Paolo Cubiotti |
| author_sort | Paolo Cubiotti |
| collection | DOAJ |
| description | LetT > 0 and Y ⊆ R^n. Given a function f:[0,T]×R^n×Y → R,we consider the Cauchy problem f(t,u,u′) = 0 in [0,T], u(0) = ξ . We prove an existence and qualitative result for the generalized solutions of the above problem. In particular, our result does not require the continuity of f with respect to the first two variables. As a matter of fact, a function f(t,x,y) satisfying our assumptions could be discontinuous (with respect to x) even at all points x ∈ R^n . We also study the dependence of the solution set S_T(ξ) from the initial point ξ ∈ R^n. In particular, we prove that, under our assumptions, the multifunction S_T admits a multivalued selection Φ which is upper semicontinuous with nonempty compact acyclic values. |
| first_indexed | 2024-04-13T13:56:30Z |
| format | Article |
| id | doaj.art-43af52d384e8495d9b8e11c0d752a048 |
| institution | Directory Open Access Journal |
| issn | 0365-0359 1825-1242 |
| language | English |
| last_indexed | 2024-04-13T13:56:30Z |
| publishDate | 2018-10-01 |
| publisher | Accademia Peloritana dei Pericolanti |
| record_format | Article |
| series | Atti della Accademia Peloritana dei Pericolanti : Classe di Scienze Fisiche, Matematiche e Naturali |
| spelling | doaj.art-43af52d384e8495d9b8e11c0d752a0482022-12-22T02:44:11ZengAccademia Peloritana dei PericolantiAtti della Accademia Peloritana dei Pericolanti : Classe di Scienze Fisiche, Matematiche e Naturali0365-03591825-12422018-10-01962A410.1478/AAPP.962A4AAPP.962A4An existence and qualitative result for discontinuous implicit differential equationsPaolo CubiottiLetT > 0 and Y ⊆ R^n. Given a function f:[0,T]×R^n×Y → R,we consider the Cauchy problem f(t,u,u′) = 0 in [0,T], u(0) = ξ . We prove an existence and qualitative result for the generalized solutions of the above problem. In particular, our result does not require the continuity of f with respect to the first two variables. As a matter of fact, a function f(t,x,y) satisfying our assumptions could be discontinuous (with respect to x) even at all points x ∈ R^n . We also study the dependence of the solution set S_T(ξ) from the initial point ξ ∈ R^n. In particular, we prove that, under our assumptions, the multifunction S_T admits a multivalued selection Φ which is upper semicontinuous with nonempty compact acyclic values. http://dx.doi.org/10.1478/AAPP.962A4 |
| spellingShingle | Paolo Cubiotti An existence and qualitative result for discontinuous implicit differential equations Atti della Accademia Peloritana dei Pericolanti : Classe di Scienze Fisiche, Matematiche e Naturali |
| title | An existence and qualitative result for discontinuous implicit differential equations |
| title_full | An existence and qualitative result for discontinuous implicit differential equations |
| title_fullStr | An existence and qualitative result for discontinuous implicit differential equations |
| title_full_unstemmed | An existence and qualitative result for discontinuous implicit differential equations |
| title_short | An existence and qualitative result for discontinuous implicit differential equations |
| title_sort | existence and qualitative result for discontinuous implicit differential equations |
| url |
http://dx.doi.org/10.1478/AAPP.962A4
|
| work_keys_str_mv | AT paolocubiotti anexistenceandqualitativeresultfordiscontinuousimplicitdifferentialequations AT paolocubiotti existenceandqualitativeresultfordiscontinuousimplicitdifferentialequations |