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.
Highlights
Received: 1 December 2021The theory of differential equations involving the Stieltjes derivative [1] instead of the usual derivative has recently developed significantly
Aware of the significance of periodic boundary conditions when studying real life processes, the present work focuses on the analysis of a first-order differential inclusion with periodic boundary value conditions
We prove that our Theorem 3 yields an existence result for dynamic inclusions on time scales with periodic boundary conditions
Summary
The theory of differential equations involving the Stieltjes derivative [1] instead of the usual derivative has recently developed significantly (we refer to [2,3,4] and the references therein). Mathematics 2022, 10, 55 involving the Stieltjes derivative with respect to a left-continuous non-decreasing function g : [0, T ] → R and a real function p, Lebesgue-Stieltjes integrable with respect to g This problem was investigated in the single-valued framework in [16] and generalized to the multivalued setting in [17] under the assumptions that F : [0, T ] × Rd → P (Rd ). Applying an idea presented (and used in the study of initial value problems) in [4], we prove that the existence results in [17] can be obtained in the less restrictive case where F is convex, compact-valued, and upper semi-continuous with respect to its second variable, except for a set (which can be dense in Rd , e.g., Example 3.11 in [4]), where a condition involving the notion of contingent g-derivative must be imposed. We generalize several results in the literature, such as the multivalued results in [10,26,27] (or the single-valued results in [28,29]) where the multifunction on the right-hand side was supposed to be convex, compact-valued, and upper semicontinuous concerning the second argument
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