Abstract
In the present paper, we are interested in studying first-order Stieltjes differential inclusions with periodic boundary conditions. Relying on recent results obtained by the authors in the single-valued case, the existence of regulated solutions is obtained via the multivalued Bohnenblust–Karlin fixed-point theorem and a result concerning the dependence on the data of the solution set is provided.
Highlights
Allowing the study in a unique framework of many classical problems: ordinary differential or difference equations, impulsive differential problems, dynamic equations on time scales and generalized differential equations (e.g., [2,3]), it is clear why the theory of differential equations driven by measures has seen a significant growth (e.g., [1,4])
Based on the results obtained in [4] for measure-driven differential equations with periodic boundary conditions, in the present paper we focus on nonlinear differential inclusions of the form: (
In the particular case of the identical function g, periodic differential problems have been widely considered in the literature; to mention only a few works, we refer to [16,17,18] for the single-valued setting and to [19,20] or [21,22] in the set-valued framework
Summary
Allowing the study in a unique framework of many classical problems: ordinary differential or difference equations (in the case of an absolutely continuous measure—with respect to the Lebesgue measure—respectively of a discrete measure), impulsive differential problems (for a sum of Lebesgue measure with a discrete one), dynamic equations on time scales (see [1]) and generalized differential equations (e.g., [2,3]), it is clear why the theory of differential equations driven by measures has seen a significant growth (e.g., [1,4]). Based on the results obtained in [4] for measure-driven differential equations with periodic boundary conditions, in the present paper we focus on nonlinear differential inclusions of the form:. As far as the authors know, periodic differential problems driven by a non-decreasing left-continuous function g have been studied only in the single-valued case in [4]. Having in mind that the theory of measure-driven equations is equivalent, in most situations, with the theory of dynamic equations on time scales ([1], see [23]), our study could be used to deduce new existence and dependence on the data results for periodic dynamic inclusions on time scales (see [24,25]). After introducing the notations and recalling some necessary known facts, in Section 3 we present an existence result for the single-valued case and we proceed to the main results in Section 4: we prove (for the multivalued setting) an existence result and a result on the dependence of the solution set on the data
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