Abstract
The paper deals with semilinear differential inclusions with state-dependent impulses in Banach spaces. Defining a suitable Banach space in which all the solutions can be embedded we prove the first existence result for at least one global mild solution of the problem considered. Then we characterize this result by means of a new definition of Lyapunov pairs. To this aim we give sufficient conditions for the existence of Lyapunov pairs in terms of a new concept of contingent derivative.
Highlights
Impulsive differential equations or inclusions describe phenomena characterized by the fact that the model parameters are subject to short-term perturbations in time
Rubbioni paola.rubbioni@unipg.it 1 Department of Mathematics and Computer Science, University of Perugia, Perugia, Italy 2 Faculty of Mathematics and Computer Science, Nicolaus Copernicus University in Torun, Torun, Poland the periodic treatment of some diseases, impulses may correspond to administration of a drug treatment; in environmental sciences, impulses may correspond to seasonal changes or harvesting; in economics, impulses may correspond to abrupt changes of prices
For a bibliography on the theory of impulsive differential equations one can see, for instance, the monographs [4, 25] and for more recent results on impulsive differential inclusions we refer the interested reader to some papers and monographs of the last decade: [7, 8, 18, 19]
Summary
Impulsive differential equations or inclusions describe phenomena characterized by the fact that the model parameters are subject to short-term perturbations in time. Assuming that A is a linear operator generating a compact semigroup, Carjain [10] extends the results obtained in the previous cited papers to the case of a multivalued map F To this aim he introduces a new concept of contingent derivative suitable for inclusions. The paper is organized as follows: in Section 2 we prove an existence result for the impulsive problem (I P ); in Section 3, after a preliminary study of the autonomous case (see Section 3.1), we give sufficient conditions to have an equi-Lyapunov pair for the differential inclusion (2) (see Section 3.2) and we obtain another existence result for (I P ) via equi-Lyapunov pairs defined as in Definition 1.5 (see Section 3.3); in Section 4 we give the concluding remarks; for the reader’s convenience in the Appendix we recall some definitions and results the proofs are based on
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: 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.
