Abstract
Let T > 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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
More From: Atti della Accademia Peloritana dei Pericolanti : Classe di Scienze Fisiche, Matematiche e Naturali
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.