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On Some Properties of a Class of Eventually Locally Mixed Cyclic/Acyclic Multivalued Self-Mappings with Application Examples

Author

Listed:
  • Manuel De la Sen

    (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, Bizkaia, Spain)

  • Asier Ibeas

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

Abstract

In this paper, a multivalued self-mapping is defined on the union of a finite number of subsets p ≥ 2 of a metric space which is, in general, of a mixed cyclic and acyclic nature in the sense that it can perform some iterations within each of the subsets before executing a switching action to its right adjacent one when generating orbits. The self-mapping can have combinations of locally contractive, non-contractive/non-expansive and locally expansive properties for some of the switching between different pairs of adjacent subsets. The properties of the asymptotic boundedness of the distances associated with the elements of the orbits are achieved under certain conditions of the global dominance of the contractivity of groups of consecutive iterations of the self-mapping, with each of those groups being of non-necessarily fixed size. If the metric space is a uniformly convex Banach one and the subsets are closed and convex, then some particular results on the convergence of the sequences of iterates to the best proximity points of the adjacent subsets are obtained in the absence of eventual local expansivity for switches between all the pairs of adjacent subsets. An application of the stabilization of a discrete dynamic system subject to impulsive effects in its dynamics due to finite discontinuity jumps in its state is also discussed.

Suggested Citation

  • Manuel De la Sen & Asier Ibeas, 2022. "On Some Properties of a Class of Eventually Locally Mixed Cyclic/Acyclic Multivalued Self-Mappings with Application Examples," Mathematics, MDPI, vol. 10(14), pages 1-29, July.
  • Handle: RePEc:gam:jmathe:v:10:y:2022:i:14:p:2415-:d:860019
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    References listed on IDEAS

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    1. Victoria Olisama & Johnson Olaleru & Hudson Akewe, 2017. "Best Proximity Point Results for Some Contractive Mappings in Uniform Spaces," International Journal of Analysis, Hindawi, vol. 2017, pages 1-8, April.
    2. Ali Abkar & Moosa Gabeleh, 2013. "Best proximity points of non-self mappings," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(2), pages 287-295, July.
    3. Zhu, Xingao & Liu, Shutang, 2022. "Reachable set estimation for continuous-time impulsive switched nonlinear time-varying systems with delay and disturbance," Applied Mathematics and Computation, Elsevier, vol. 420(C).
    4. A. Amini-Harandi, 2013. "Best proximity point theorems for cyclic strongly quasi-contraction mappings," Journal of Global Optimization, Springer, vol. 56(4), pages 1667-1674, August.
    5. Jin-E Zhang & Xiangru Xing & Jinliang Wang, 2022. "Stabilization of Uncertain Switched Systems with Frequent Asynchronism via Event-Triggered Dynamic Output-Feedback Control," Discrete Dynamics in Nature and Society, Hindawi, vol. 2022, pages 1-18, April.
    6. M. de la Sen & A. Ibeas, 2008. "Stability Results for Switched Linear Systems with Constant Discrete Delays," Mathematical Problems in Engineering, Hindawi, vol. 2008, pages 1-28, March.
    7. Pragati Gautam & Shiv Raj Singh & Santosh Kumar & Swapnil Verma & Naeem Jan, 2022. "On Nonunique Fixed Point Theorems via Interpolative Chatterjea Type Suzuki Contraction in Quasi-Partial b-Metric Space," Journal of Mathematics, Hindawi, vol. 2022, pages 1-10, April.
    8. Fei Meng & Xuyu Shen & Xiaofeng Li, 2022. "Stability Analysis and Synthesis for 2-D Switched Systems with Random Disturbance," Mathematics, MDPI, vol. 10(5), pages 1-18, March.
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