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Fixed Points of Self-Mappings with Jumping Effects: Application to Stability of a Class of Impulsive Dynamic Systems

Author

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  • Manuel De la Sen

    (Automatic Control Group–ACG, Institute of Research and Development of Processes, Department of Electricity and Electronics, Faculty of Science and Technology, University of the Basque Country (UPV/EHU), 48940 Leioa, Spain)

  • Asier Ibeas

    (Department of Telecommunications and Systems Engineering, Universitat Autònoma de Barcelona, UAB, 08193 Barcelona, Spain)

  • Aitor J. Garrido

    (Automatic Control Group–ACG, Institute of Research and Development of Processes, Department of Automatic Control and Systems Engineering, Faculty of Engineering of Bilbao, Institute of Research and Development of Processes–IIDP, University of the Basque Country (UPV/EHU), 48013 Bilbao, Spain)

  • Izaskun Garrido

    (Automatic Control Group–ACG, Institute of Research and Development of Processes, Department of Automatic Control and Systems Engineering, Faculty of Engineering of Bilbao, Institute of Research and Development of Processes–IIDP, University of the Basque Country (UPV/EHU), 48013 Bilbao, Spain)

Abstract

This paper studies the boundedness and convergence properties of the sequences generated by strict and weak contractions in metric spaces, as well as their fixed points, in the event that finite jumps can take place from the left to the right limits of the successive values of the generated sequences. An application is devoted to the stabilization and the asymptotic stabilization of impulsive linear time-varying dynamic systems of the n -th order. The impulses are formalized based on the theory of Dirac distributions. Several results are stated and proved, namely, (a) for the case when the time derivative of the differential system is impulsive at isolated time instants; (b) for the case when the matrix function of dynamics is almost everywhere differentiable with impulsive effects at isolated time instants; and (c) for the case of combinations of the two above effects, which can either jointly take place at the same time instants or at distinct time instants. In the first case, finite discontinuities of the first order in the solution are generated; that is, equivalently, finite jumps take place between the corresponding left and right limits of the solution at the impulsive time instants. The second case generates, equivalently, finite jumps in the first derivative of the solution with respect to time from their left to their right limits at the corresponding impulsive time instants. Finally, the third case exhibits both of the above effects in a combined way.

Suggested Citation

  • Manuel De la Sen & Asier Ibeas & Aitor J. Garrido & Izaskun Garrido, 2025. "Fixed Points of Self-Mappings with Jumping Effects: Application to Stability of a Class of Impulsive Dynamic Systems," Mathematics, MDPI, vol. 13(7), pages 1-42, March.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:7:p:1157-:d:1625190
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    References listed on IDEAS

    as
    1. Nguyen Huu Sau & Mai Viet Thuan & Nguyen Thi Phuong, 2024. "Exponential stability for discrete-time impulsive positive singular system with time delays," International Journal of Systems Science, Taylor & Francis Journals, vol. 55(8), pages 1510-1527, June.
    2. Erdal Karapınar & Andreea Fulga & Maliha Rashid & Lariab Shahid & Hassen Aydi, 2019. "Large Contractions on Quasi-Metric Spaces with an Application to Nonlinear Fractional Differential Equations," Mathematics, MDPI, vol. 7(5), pages 1-11, May.
    3. Ebrahem Ateatullah Algehyne & Musaad Sabih Aldhabani & Faizan Ahmad Khan, 2023. "Relational Contractions Involving (c)-Comparison Functions with Applications to Boundary Value Problems," Mathematics, MDPI, vol. 11(6), pages 1-11, March.
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