Abstract
Existence results for a Cauchy problem driven by a semilinear differential Sturm-Liouville inclusion are achived by proving, in a preliminary way, an existence theorem for a suitable integral inclusion. In order to obtain this proposition we use a recent fixed point theorem that allows us to work with the weak topology and the De Blasi measure of weak noncompactness. So we avoid requests of compactness on the multivalued terms. Then, by requiring different properties on the map p involved in the Sturm-Liouville inclusion, we are able to establish the existence of both mild solutions and strong ones for the problem examinated. Moreover we focus our attention on the study of controllability for a Cauchy problem governed by a suitable Sturm-Liouville equation. Finally we precise that our results are able to study problems involving a more general version of a semilinear differential Sturm-Liouville inclusion.
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Open access funding provided by Università degli Studi di Perugia within the CRUI-CARE Agreement. This research is carried out within the national group GNAMPA of INdAM.The first author is partially supported by the Department of Mathematics and Computer Science of the University of Perugia (Italy) and by the projects “Fondi di funzionamento per la ricerca dipartimentale-Anno 2021”, “Metodi della Teoria dell’Approssimazione, Analisi Reale, Analisi Nonlineare e loro applicazioni”, “Integrazione, Approssimazione, Analisi Nonlineare e loro Applicazioni”, funded by the Fund for Basic Research 2018 and 2019 of the University of Perugia. The authors have not disclosed any funding.
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Writing-original draft preparation, T.C. and G.D. All authors contributed equally and singnificantly in writing this article. All authors read and approved the final manuscript.
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Cardinali, T., Duricchi, G. Strong Solutions and Mild Solutions for Sturm-Liouville Differential Inclusions. Set-Valued Var. Anal 32, 3 (2024). https://doi.org/10.1007/s11228-024-00706-6
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DOI: https://doi.org/10.1007/s11228-024-00706-6
Keywords
- Sturm-Liouville differential inclusions
- Integral inclusions
- Radon-Nikodym property
- Measure of weak noncompactness
- Controllability
- Fixed point theorem