Stieltjes Differential Inclusions with Periodic Boundary Conditions without Upper Semicontinuity
We are studying first order differential inclusions with periodic boundary conditions where the Stieltjes derivative with respect to a left-continuous non-decreasing function replaces the classical derivative. The involved set-valued mapping is not assumed to have compact and convex values, nor to b...
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MDPI AG
2021-12-01
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| author | Valeria Marraffa Bianca Satco |
| author_facet | Valeria Marraffa Bianca Satco |
| author_sort | Valeria Marraffa |
| collection | DOAJ |
| description | We are studying first order differential inclusions with periodic boundary conditions where the Stieltjes derivative with respect to a left-continuous non-decreasing function replaces the classical derivative. The involved set-valued mapping is not assumed to have compact and convex values, nor to be upper semicontinuous concerning the second argument everywhere, as in other related works. A condition involving the contingent derivative relative to the non-decreasing function (recently introduced and applied to initial value problems by R.L. Pouso, I.M. Marquez Albes, and J. Rodriguez-Lopez) is imposed on the set where the upper semicontinuity and the assumption to have compact convex values fail. Based on previously obtained results for periodic problems in the single-valued cases, the existence of solutions is proven. It is also pointed out that the solution set is compact in the uniform convergence topology. In particular, the existence results are obtained for periodic impulsive differential inclusions (with multivalued impulsive maps and finite or possibly countable impulsive moments) without upper semicontinuity assumptions on the right-hand side, and also the existence of solutions is derived for dynamic inclusions on time scales with periodic boundary conditions. |
| first_indexed | 2024-03-10T03:31:33Z |
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| language | English |
| last_indexed | 2024-03-10T03:31:33Z |
| publishDate | 2021-12-01 |
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| spelling | doaj.art-193154af35ff4bd9a919ce29086f21092023-11-23T11:53:26ZengMDPI AGMathematics2227-73902021-12-011015510.3390/math10010055Stieltjes Differential Inclusions with Periodic Boundary Conditions without Upper SemicontinuityValeria Marraffa0Bianca Satco1Dipartimento di Matematica e Informatica, Università degli Studi di Palermo, Via Archirafi 34, 90123 Palermo, ItalyFaculty of Electrical Engineering and Computer Science, Stefan cel Mare University of Suceava, Universitatii 13, 720225 Suceava, RomaniaWe are studying first order differential inclusions with periodic boundary conditions where the Stieltjes derivative with respect to a left-continuous non-decreasing function replaces the classical derivative. The involved set-valued mapping is not assumed to have compact and convex values, nor to be upper semicontinuous concerning the second argument everywhere, as in other related works. A condition involving the contingent derivative relative to the non-decreasing function (recently introduced and applied to initial value problems by R.L. Pouso, I.M. Marquez Albes, and J. Rodriguez-Lopez) is imposed on the set where the upper semicontinuity and the assumption to have compact convex values fail. Based on previously obtained results for periodic problems in the single-valued cases, the existence of solutions is proven. It is also pointed out that the solution set is compact in the uniform convergence topology. In particular, the existence results are obtained for periodic impulsive differential inclusions (with multivalued impulsive maps and finite or possibly countable impulsive moments) without upper semicontinuity assumptions on the right-hand side, and also the existence of solutions is derived for dynamic inclusions on time scales with periodic boundary conditions.https://www.mdpi.com/2227-7390/10/1/55differential inclusionperiodic boundary value conditionStieltjes derivativeimpulsedynamic equation on time scales |
| spellingShingle | Valeria Marraffa Bianca Satco Stieltjes Differential Inclusions with Periodic Boundary Conditions without Upper Semicontinuity Mathematics differential inclusion periodic boundary value condition Stieltjes derivative impulse dynamic equation on time scales |
| title | Stieltjes Differential Inclusions with Periodic Boundary Conditions without Upper Semicontinuity |
| title_full | Stieltjes Differential Inclusions with Periodic Boundary Conditions without Upper Semicontinuity |
| title_fullStr | Stieltjes Differential Inclusions with Periodic Boundary Conditions without Upper Semicontinuity |
| title_full_unstemmed | Stieltjes Differential Inclusions with Periodic Boundary Conditions without Upper Semicontinuity |
| title_short | Stieltjes Differential Inclusions with Periodic Boundary Conditions without Upper Semicontinuity |
| title_sort | stieltjes differential inclusions with periodic boundary conditions without upper semicontinuity |
| topic | differential inclusion periodic boundary value condition Stieltjes derivative impulse dynamic equation on time scales |
| url | https://www.mdpi.com/2227-7390/10/1/55 |
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