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Measure Differential Inclusions Through Selection Principles in the Space of Regulated Functions

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Abstract

Using a characterization of the space of regulated functions on a compact interval [0, 1] endowed with the topology of uniform convergence as being isometrically isomorphic to some space of continuous functions, we get several theorems on the existence of regulated selections for multifunctions and we relate them with other known selection results. In particular, we prove a version of Michael’s selection theorem for regulated functions and we prove that regulated multifunctions (with respect to the Pompeiu–Hausdorff distance) possess regulated selections. As an application, the existence of regulated solutions for a very general measure differential inclusion is presented. Having in mind that impulsive differential problems, generalized differential equations or dynamic problems on time scales can be seen as measure differential problems, the announced result possesses a high degree of generality.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, A. Mickiewicz University, Umultowska 87, 61-614, Poznań, Poland

    Mieczysław Cichoń

  2. Institute of Mathematics, Faculty of Electrical Engineering, Poznan University of Technology, Piotrowo 3A, 60-965, Poznań, Poland

    Kinga Cichoń

  3. Faculty of Electrical Engineering and Computer Science, Integrated Center for Research, Development and Innovation in Advanced Materials, Nanotechnologies, and Distributed Systems for Fabrication and Control (MANSiD), “Stefan cel Mare” University of Suceava, Universitatii 13, Suceava, Romania

    Bianca Satco

Authors
  1. Mieczysław Cichoń
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  2. Kinga Cichoń
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  3. Bianca Satco
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Correspondence to Bianca Satco.

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Cichoń, M., Cichoń, K. & Satco, B. Measure Differential Inclusions Through Selection Principles in the Space of Regulated Functions. Mediterr. J. Math. 15, 148 (2018). https://doi.org/10.1007/s00009-018-1192-y

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  • DOI: https://doi.org/10.1007/s00009-018-1192-y

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