Stieltjes Differential Inclusions with Periodic Boundary Conditions without Upper Semicontinuity

Valeria Marraffa and Bianca Satco
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Valeria Marraffa: Dipartimento di Matematica e Informatica, Università degli Studi di Palermo, Via Archirafi 34, 90123 Palermo, Italy
Bianca Satco: Faculty of Electrical Engineering and Computer Science, Stefan cel Mare University of Suceava, Universitatii 13, 720225 Suceava, Romania

Mathematics, 2021, vol. 10, issue 1, 1-17

Abstract: 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.

Keywords: differential inclusion; periodic boundary value condition; Stieltjes derivative; impulse; dynamic equation on time scales (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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