Skip to main content
SpringerLink
Log in

Coincidence Points of Two Mappings Acting from a Partially Ordered Space to an Arbitrary Set

  • Published:
Russian Mathematics Aims and scope Submit manuscript

Abstract

A coincidence point of a pair of mappings is an element, at which these mappings take on the same value. Coincidence points of mappings of partially ordered spaces were studied by A.V. Arutyunov, E.S. Zhukovskiy, and S.E. Zhukovskiy (see Topology and its Applications, 2015, V. 179, No. 1, p. 13–33); they proved, in particular, that an orderly covering mapping and a monotone one, both acting from a partially ordered space to a partially ordered one, possess a coincidence point. In this paper, we study the existence of a coincidence point of a pair of mappings that act from a partially ordered space to a set, where no binary relation is defined, and, consequently, it is impossible to define covering and monotonicity properties of mappings. In order to study the mentioned problem, we define the notion of a “quasi-coincidence” point. We understand it as an element, for which there exists another element, which does not exceed the initial one and is such that the value of the first mapping at it equals the value of the second mapping at the initial element. It turns out that the following condition is sufficient for the existence of a coincidence point: any chain of “quasi-coincidence” points is bounded and has a lower bound, which also is a “quasi-coincidence” point. We give an example of mappings that satisfy the above requirements and do not allow the application of results obtained for coincidence points of orderly covering and monotone mappings. In addition, we give an interpretation of the stability notion for coincidence points of mappings that act in partially ordered spaces with respect to their small perturbations and establish the corresponding stability conditions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arutyunov, A.V., Zhukovskiy, E.S., Zhukovskiy, S.E. “Coincidence Points Principle for Mappings in Partially Ordered Spaces”, Topology and its Appl. 179 (1), 13–33 (2015).

    Article  MathSciNet  Google Scholar 

  2. Arutyunov, A.V., Zhukovskiy, E.S., Zhukovskiy S.E. “Coincidence Points Principle for Set-Valued Mappings in Partially Ordered Spaces”, Topology and its Appl. 201, 330–343 (2016).

    Article  MathSciNet  Google Scholar 

  3. Arutyunov, A.V., Zhukovskiy, E.S., Zhukovskiy S.E. “Coincidence Points of Mappings in Partially Ordered Spaces”, Dokl. Ross. Akad. Nauk 453 (5), 475–478 (2013).

    MathSciNet  MATH  Google Scholar 

  4. Arutyunov, A.V., Zhukovskiy, E.S., Zhukovskiy S.E. “Coincidence Points Principle for Set-Valued Mappings in Partially Ordered Spaces”, Dokl. Ross. Akad. Nauk 453 (6), 595–598 (2013).

    MATH  Google Scholar 

  5. Arutyunov, A.V. “Covering Mappings in Metric Spaces and Fixed Points”, Dokl. Ross. Akad. Nauk 416 (2), 151–155 (2007).

    MathSciNet  Google Scholar 

  6. Zhukovskiy, E.S. “On Ordered-Covering Mappings and Implicit Differential Inequalities”, Differential Equations 52, 1539–1556 (2016).

    Article  MathSciNet  Google Scholar 

  7. Zhukovskiy, E.S. “On Order Covering Maps in Ordered Spaces and Chaplygin-Type Inequalities”, Algebra i Analiz, 30 (1), 96–127 (2018).

    MathSciNet  Google Scholar 

  8. Arutyunov, A.V., Zhukovskiy, E.S., Zhukovskiy S.E. “Caristi-Like Condition and the Existence of Minima of Mappings in Partially Ordered Spaces”, Journal of Optimization Theory and Appl. 180 (1), 48–61 (2018).

    Article  MathSciNet  Google Scholar 

  9. Benarab, S., Zhukovskiy, E.S. “On the Conditions of Existence of Coincidence Points for Mappings in Partially Ordered Spaces”, Vestn. Tambovsk. Un-ta. Ser. Estestvenn. i Tekjnicheskie Nauki. Tambov 23 (121), 10–16 (2018).

    Google Scholar 

  10. Benarab, S., Zhukovskiy, E.S. “About Covering Mappings with Values in the Space with a Reflexive Binary Relation”, Vestn. Tambovsk. Un-ta. Ser. Estestvenn. i Tekjnicheskie Nauki. Tambov 23 (122), 210–215 (2018).

    Google Scholar 

Download references

Funding

This work was supported by the Russian Foundation for Basic Research (project no. 17-51-12064). Results described in Section 4 were obtained by the second author with the support of the Russian Science Foundation (project no. 20-11-20131).

Author information

Authors and Affiliations

  1. Derzhavin Tambov State University, 33 Internatsionalnaya str., Tambov, 392000, Russia

    S. Benarab

  2. Derzhavin Tambov State University, 33 Internatsionalnaya str., Tambov, 392000, Russia

    E. S. Zhukovskiy

  3. V. A. Trapeznikov Institute of Control Sciences of RAS, 65 Profsoyuznaya str., Moscow, 117997, Russia

    E. S. Zhukovskiy

Authors
  1. S. Benarab
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. S. Zhukovskiy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to S. Benarab or E. S. Zhukovskiy.

Additional information

Russian Text © The Author(s), 2020, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2020, No. 5, pp. 12–21.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Benarab, S., Zhukovskiy, E.S. Coincidence Points of Two Mappings Acting from a Partially Ordered Space to an Arbitrary Set. Russ Math. 64, 8–16 (2020). https://doi.org/10.3103/S1066369X20050023

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066369X20050023

Key words

Search

Navigation