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Kantorovich’s Fixed Point Theorem and Coincidence Point Theorems for Mappings in Vector Metric Spaces

Set-Valued and Variational Analysis volume 30, pages 397–423 (2022)Cite this article

Abstract

The fixed points and coincidence points of mappings of v-metric spaces, i.e., sets on which a vector metric is defined, are investigated. The values of such a metric are elements of a cone of a Banach space rather than real nonnegative numbers. Analogs of Kantorovich’s theorems on the existence and uniqueness of a fixed point and the convergence of an iteration sequence to this point are obtained. Conditions for the existence of a coincidence point of two mappings are obtained. The results obtained are generalized to the case of set-valued mappings.

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Authors and Affiliations

  1. V.A. Trapeznikov Institute of Control Sciences of RAS, Profsoyuznaya st., 65, 117997, Moscow, Russia

    Aram V. Arutyunov, Evgeny S. Zhukovskiy, Sergey E. Zhukovskiy & Zukhra T. Zhukovskaya

  2. Derzhavin Tambov State University, Internatsional’naya str., 33, 392622, Tambov, Russia

    Evgeny S. Zhukovskiy

Authors
  1. Aram V. Arutyunov
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  2. Evgeny S. Zhukovskiy
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  3. Sergey E. Zhukovskiy
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  4. Zukhra T. Zhukovskaya
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Correspondence to Sergey E. Zhukovskiy.

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This research was supported by the grants of the Russian Foundation for Basic Research (Project no. 19-01-00080, 20-31-70013) and by the grant of the President of the Russian Federation (project no. MD-2658.2021.1.1). The results of the section 5are due to the second author who was supported by the grant of the Russian Science Foundation (Project no. 20-11-20131).

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Arutyunov, A.V., Zhukovskiy, E.S., Zhukovskiy, S.E. et al. Kantorovich’s Fixed Point Theorem and Coincidence Point Theorems for Mappings in Vector Metric Spaces. Set-Valued Var. Anal 30, 397–423 (2022). https://doi.org/10.1007/s11228-021-00588-y

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  • DOI: https://doi.org/10.1007/s11228-021-00588-y

Keywords

  • Kantorovich’s fixed point theorem
  • Coincidence point
  • Vector metric space

Mathematics Subject Classification (2010)

  • 49J53
  • 54H25