Further study on second order nonlocal problems monitored by an operator: an approach without compactness
In this note we prove the existence of mild solutions for nonlocal problems governed by semilinear second order differential inclusions which involves a nonlinear term driven by an operator. A first result is obtained in suitable Banach spaces in the lack of compactness both on the fundamental opera...
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| Format: | Article |
| Language: | English |
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University of Szeged
2023-05-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=10230 |
| _version_ | 1797352346644119552 |
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| author | Tiziana Cardinali Giulia Duricchi |
| author_facet | Tiziana Cardinali Giulia Duricchi |
| author_sort | Tiziana Cardinali |
| collection | DOAJ |
| description | In this note we prove the existence of mild solutions for nonlocal problems governed by semilinear second order differential inclusions which involves a nonlinear term driven by an operator. A first result is obtained in suitable Banach spaces in the lack of compactness both on the fundamental operator, generated by the linear part, and on the nonlinear multivalued term. This purpose is achieved by combining a fixed point theorem, a selection theorem and a containment theorem. Further we provide another existence result in reflexive spaces by using the classical Hahn–Banach theorem and a new selection proposition, proved here, for a multimap guided by an operator. This setting allows us to remove some assumptions required in the previous existence theorem. As a consequence of this last result we obtain the controllability of a problem driven by a wave equation on which an appropriate perturbation acts. |
| first_indexed | 2024-03-08T13:15:06Z |
| format | Article |
| id | doaj.art-30fa3153799840ad8963d9442db39861 |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-03-08T13:15:06Z |
| publishDate | 2023-05-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-30fa3153799840ad8963d9442db398612024-01-18T08:28:07ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752023-05-0120231313410.14232/ejqtde.2023.1.1310230Further study on second order nonlocal problems monitored by an operator: an approach without compactnessTiziana Cardinali0Giulia Duricchi1Dept. of Mathematics and Computer Sciences, University of Perugia, ItalyDept. of Mathematics, University of Firenze, ItalyIn this note we prove the existence of mild solutions for nonlocal problems governed by semilinear second order differential inclusions which involves a nonlinear term driven by an operator. A first result is obtained in suitable Banach spaces in the lack of compactness both on the fundamental operator, generated by the linear part, and on the nonlinear multivalued term. This purpose is achieved by combining a fixed point theorem, a selection theorem and a containment theorem. Further we provide another existence result in reflexive spaces by using the classical Hahn–Banach theorem and a new selection proposition, proved here, for a multimap guided by an operator. This setting allows us to remove some assumptions required in the previous existence theorem. As a consequence of this last result we obtain the controllability of a problem driven by a wave equation on which an appropriate perturbation acts.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=10230semilinear second order differential inclusionperturbation effectfundamental systemde blasi measure of noncompactnesscontrollability problemwave equation |
| spellingShingle | Tiziana Cardinali Giulia Duricchi Further study on second order nonlocal problems monitored by an operator: an approach without compactness Electronic Journal of Qualitative Theory of Differential Equations semilinear second order differential inclusion perturbation effect fundamental system de blasi measure of noncompactness controllability problem wave equation |
| title | Further study on second order nonlocal problems monitored by an operator: an approach without compactness |
| title_full | Further study on second order nonlocal problems monitored by an operator: an approach without compactness |
| title_fullStr | Further study on second order nonlocal problems monitored by an operator: an approach without compactness |
| title_full_unstemmed | Further study on second order nonlocal problems monitored by an operator: an approach without compactness |
| title_short | Further study on second order nonlocal problems monitored by an operator: an approach without compactness |
| title_sort | further study on second order nonlocal problems monitored by an operator an approach without compactness |
| topic | semilinear second order differential inclusion perturbation effect fundamental system de blasi measure of noncompactness controllability problem wave equation |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=10230 |
| work_keys_str_mv | AT tizianacardinali furtherstudyonsecondordernonlocalproblemsmonitoredbyanoperatoranapproachwithoutcompactness AT giuliaduricchi furtherstudyonsecondordernonlocalproblemsmonitoredbyanoperatoranapproachwithoutcompactness |